Proof. We prove this using the Yoneda lemma. Let $m \in \mathcal {M}$. Recall that $F\colon \mathcal {C} \to \mathcal {D}$ is a map in $\mathrm {Mod}_\mathcal {M}$, so that it commutes with $m \otimes -$, so we conclude that

\begin{align*} \hom(m, \hom^\mathcal{M}(FX, Y)) &\cong \hom(m \otimes FX, Y)\\ &\cong \hom(F(m \otimes X), Y)\\ &\cong \hom(m \otimes X, GY)\\ &\cong \hom(m, \hom^\mathcal{M}(X, GY)) . \end{align*}

The fact that it is a lift of the $\mathcal {S}$-enriched hom is the case $m = \mathbf {1}_\mathcal {M}$.

Proof. Let $X \in \mathcal {S}^{\mathrm {at}}$ be atomic, then for any $Y \in \mathcal {S}$ we have

\[ \hom(X, Y) \cong \hom(X, \operatorname*{colim}_Y *) \cong \operatorname*{colim}_Y \hom(X, *) \cong \operatorname*{colim}_Y * \cong Y . \]

Thus, $X$ corepresents the identity functor $\mathrm {id}\colon \mathcal {S} \to \mathcal {S}$, namely $X = *$.

Proof. Let $X \in \mathcal {C}$. First assume that $X$ is atomic. Recall that $\Omega ^\infty \colon \mathrm {Sp} \to \mathcal {S}$ commutes with filtered colimits, so that $\hom (X, -) \cong \Omega ^\infty \hom ^\mathrm {Sp}(X, -)$ commutes with filtered colimits, i.e. it is compact.

Now assume that $X$ is compact. Recall that for any $n \in \mathbb {Z}$, the functor $\Sigma ^n\colon \mathrm {Sp} \to \mathrm {Sp}$ commutes with all limits and colimits and in particular with filtered colimits, thus $\hom (X, \Sigma ^n -) \cong \Omega ^\infty \Sigma ^n \hom ^\mathrm {Sp}(X, -)$ also commutes with filtered colimits. In addition, the functors $\Omega ^\infty \Sigma ^n\colon \mathrm {Sp} \to \mathcal {S}$ are jointly conservative, implying that $\hom ^\mathrm {Sp}(X, -)$ commutes with filtered colimits. Furthermore, it commutes with finite limits, thus by stability also with all finite colimits, which together with filtered colimits generate all colimits.

Proof. Let $\kappa$ be a regular cardinal such that the unit $\mathbf {1}_\mathcal {M} \in \mathcal {M}$ is $\kappa$-compact. We show that $\mathcal {C}^{\mathrm {at}} \subseteq \mathcal {C}^\kappa$, that is the atomics are $\kappa$-compact. Let $X \in \mathcal {C}$ be an atomic object, so in particular $\hom ^\mathcal {M}(X,-)$ commutes with $\kappa$-filtered colimits. Since $\mathbf {1}_\mathcal {M}$ is $\kappa$-compact, $\hom (\mathbf {1}_\mathcal {M},-)$ commutes with $\kappa$-filtered colimits, implying that the composition $\hom (X,-) \cong \hom (\mathbf {1}_\mathcal {M}, \hom ^\mathcal {M}(X,-))$ commutes with $\kappa$-filtered colimits.

Proof. By assumption the right adjoint $G\colon \mathcal {D} \to \mathcal {C}$ is itself a left adjoint, thus preserves colimits. Let $X \in \mathcal {C}^{\mathrm {at}}$ be an atomic object, then using Lemma 2.2 $\hom ^\mathcal {M}(FX,-) \cong \hom ^\mathcal {M}(X, G-)$, which is the composition of $G$ and $\hom ^\mathcal {M}(X, -)$, both of which preserve colimits, so that $FX$ is atomic.

Proof. We wish to show that $G$, the right adjoint of $F$, is itself a left adjoint, namely that it preserves colimits. Let $Y_i\colon I \to \mathcal {D}$ be a diagram, and we wish to show that $G(\operatorname *{colim} Y_i) \cong \operatorname *{colim} G Y_i$. By the Yoneda lemma, this is equivalent to checking that for every $X \in \mathcal {C}$ we have

\[ \hom(X, G(\operatorname*{colim} Y_i)) \cong \hom(X, \operatorname*{colim} G Y_i). \]

Since $\hom (-, -) \cong \hom (\mathbf {1}_\mathcal {M}, \hom ^\mathcal {M}(-, -))$, it suffices to check that for every $X \in \mathcal {C}$ we have

\[ \hom^\mathcal{M}(X, G(\operatorname*{colim} Y_i)) \cong \hom^\mathcal{M}(X, \operatorname*{colim} G Y_i). \]

Let $A$ denote the collection of $X \in \mathcal {C}$ for which this condition holds, and we shall show that $A = \mathcal {C}$.

First, for every $X \in B$, we know that

\begin{align*} \hom^\mathcal{M}(X, G(\operatorname*{colim}_I Y_i)) &\cong \hom^\mathcal{M}(F X, \operatorname*{colim}_I Y_i)\\ &\cong \operatorname*{colim}_I \hom^\mathcal{M}(F X, Y_i)\\ &\cong \operatorname*{colim}_I \hom^\mathcal{M}(X, G Y_i)\\ &\cong \hom^\mathcal{M}(X, \operatorname*{colim}_I G Y_i), \end{align*}

where the first and third steps follow from Lemma 2.2, the second step follows from the assumption that $FX$ is atomic since $X \in B$ and $F$ sends $B$ to atomic objects, and the fourth step follows from the fact $X$ is atomic. Therefore, $B \subseteq A$.

Second, for every $X \in A$ and $m \in \mathcal {M}$, we know that $\hom ^\mathcal {M}(m \otimes X, -) \cong \hom ^\mathcal {M}(m, \hom ^\mathcal {M}(X, -))$ so that $m \otimes X \in A$, i.e. $A$ is closed under the action of $\mathcal {M}$.

Third, for every diagram $X_j\colon J \to \mathcal {C}$ landing in $A$, we know that $\hom ^\mathcal {M}(\operatorname *{colim}_J X_j, -) \cong \lim _{J^\mathrm {op}} \hom ^\mathcal {M}(X_j, -)$ so that $\operatorname *{colim}_J X_j \in A$, i.e. $A$ is closed under colimits.

We have shown that $B \subseteq A$ and that $A$ is closed under the action of $\mathcal {M}$ and colimits, and by assumption $B$ are atomic generators, thus $A = \mathcal {C}$ as needed.

Proof. First, by Proposition 2.13 and the fact that the $\mathbf {1}_\mathcal {M}$ is atomic, the functor indeed lands in the full subcategory $\mathcal {C}^{\mathrm {at}}$. In particular, it is also fully faithful as the restriction of an equivalence to two full subcategories. We need to show that it is essentially surjective, i.e. that if $X \in \mathcal {C}^{\mathrm {at}}$ then $- \otimes X\colon \mathcal {M} \to \mathcal {C}$ is internally left adjoint. This holds since its right adjoint is ${\hom ^\mathcal {M}(X, -)\colon\mathcal {C} \to \mathcal {M}}$, which by assumption preserves colimits.

For the last part, as $- \otimes X\colon \mathcal {M} \to \mathcal {C}$ is internally left adjoint, Proposition 2.13 implies that it sends atomic objects to atomic objects.

Proof. This is immediate from the fact that $\mathrm {Mod}_\mathcal {N} \subseteq \mathrm {Mod}_\mathcal {M}$.

Proof. By assumption, $\lim _I\colon \mathcal {C}^I \to \mathcal {C}$ commutes with colimits. Therefore, it is a map in $\mathrm {Mod}_\mathcal {M}$, so that it also commutes $m \otimes -\colon \mathcal {C} \to \mathcal {C}$ for any $m \in \mathcal {M}$.

Proof. Let $X_i\colon I^\mathrm {op} \to \mathcal {C}$ be a diagram landing in the atomics. Recall that $\hom ^\mathcal {M}(-, -)\colon \mathcal {C}^\mathrm {op} \times \mathcal {C} \to \mathcal {M}$ commutes with limits in the first coordinate, thus $\hom ^\mathcal {M}(\operatorname *{colim}_{I^\mathrm {op}} X_i, -)$ is equivalent to

\[ \mathcal{C} \xrightarrow{\Delta} \mathcal{C}^{I^\mathrm{op}} \xrightarrow{(\hom^\mathcal{M}(X_i, -))_I} \mathcal{M}^I \xrightarrow{\lim_I} \mathcal{M} . \]

The functor $\Delta$ commutes with colimits since colimits in functor categories are computed level-wise. Since each $X_i$ is atomic, each $\hom ^\mathcal {M}(X_i, -)$ commutes with colimits, and as colimits in functor categories are computed level-wise, we get that $(\hom ^\mathcal {M}(X_i, -))_I$ commutes with colimits. By assumption, $I$ is an absolute limit of $\mathcal {M}$, thus $\lim _I$ commutes with colimits. This shows that $\hom ^\mathcal {M}(\operatorname *{colim}_{I^\mathrm {op}} X_i, -)$ commutes with colimits, i.e. that $\operatorname *{colim}_{I^\mathrm {op}} X_i$ is indeed atomic.

Proof. Recall that $\mathcal {C}$ itself is stable, so the first part follows from the commutativity of finite limits and colimits in stable categories. For the second part, first note that the zero object is obviously atomic. As finite limits are absolute, the atomics are closed under finite colimits, so it suffices to show that the atomics are also closed under desuspensions. Let $X \in \mathcal {C}^{{\mathcal {M}\text {-}}\mathrm {at}}$, then indeed $\hom ^\mathcal {M}(\Sigma ^{-1} X, -) \cong \Sigma \hom ^\mathcal {M}(X, -)$ is colimit preserving, because $X$ is atomic and $\Sigma \colon \mathcal {M} \to \mathcal {M}$ is colimit preserving.

Proof. By [Reference Carmeli, Schlank and YanovskiCSY21a, Proposition 5.2.15], the localization is smashing if and only if the right adjoint of $F\colon \mathcal {M} \to \mathcal {N}$ is itself a left adjoint (i.e. if $F$ is internally left adjoint). This allows us to treat $\mathcal {N}$ as a full subcategory of $\mathcal {M}$ closed under both limits and colimits. Now, [Reference Carmeli, Schlank and YanovskiCSY21a, Proposition 5.2.10] shows that $\mathcal {N}$ classifies the property of being in the mode $\mathcal {M}$ such that all $\mathcal {M}$-enriched $\hom$'s land in $\mathcal {N}$, i.e. $\hom ^\mathcal {M}(X,-) = \hom ^\mathcal {N}(X,-)$, thus by the closure of $\mathcal {N} \subseteq \mathcal {M}$ under colimits, $\hom ^\mathcal {M}(X,-)\colon \mathcal {C} \to \mathcal {M}$ commutes with colimits if and only if $\hom ^\mathcal {N}(X,-)\colon \mathcal {C} \to \mathcal {N}$ does.

Proof. Indeed, we have an equivalence

\begin{align*} \operatorname{\operatorname{\mathcal{P}}_\mathcal{K}}(\mathcal{C}_0) \otimes \mathcal{M} &\cong \operatorname{Fun}^\mathrm{R}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}}(\mathcal{C}_0)^\mathrm{op}, \mathcal{M})\\ &\cong \operatorname{Fun}^\mathrm{L}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}}(\mathcal{C}_0), \mathcal{M}^\mathrm{op})^\mathrm{op}\\ &\cong \operatorname{Fun}_\mathcal{K}(\mathcal{C}_0, \mathcal{M}^\mathrm{op})^\mathrm{op}\\ &\cong \operatorname{Fun}^{\mathcal{K}^\mathrm{op}}(\mathcal{C}_0^\mathrm{op}, \mathcal{M})\\ &= \operatorname{\operatorname{\mathcal{P}}_\mathcal{K}^\mathcal{M}}(\mathcal{C}_0) , \end{align*}

where the first equality is [Reference LurieLur17, Proposition 4.8.1.17], the second is passing to the opposite, the third is the universal property of $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}}$ given in [Reference LurieLur09, Corollary 5.3.6.10], the fourth is by passing to the opposite, and the last is by definition.

Proof. Let $F_j\colon J \to \operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$ be a diagram landing in $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$. We need to show that $\operatorname *{colim}_J F_j$ and $\lim _J F_j$ are again in $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$, i.e. that they commute with all limits indexed by $I \in {\mathcal {K}^\mathrm {op}}$. Let $X_i\colon I^\mathrm {op} \to \mathcal {C}_0$ be a diagram. Using the fact that colimits and limits in functor categories are computed level-wise, and that $I$ is an absolute limit, we get

\[ \operatorname*{colim}_J F_j(\operatorname*{colim}_{I^\mathrm{op}} X_i) \cong \operatorname*{colim}_J \lim_I F_j(X_i) \cong \lim_I \operatorname*{colim}_J F_j(X_i) . \]

Similarly $\lim _J F_j \in \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$, since limits commute with limits.

Proof. Recall that $f^*$ is given by pre-composition with $f^\mathrm {op}\colon \mathcal {C}_0^\mathrm {op} \to \mathcal {D}_0^\mathrm {op}$, which preserves limits indexed by $I \in {\mathcal {K}^\mathrm {op}}$, as the opposite of a morphism in $\mathrm {Cat}_\mathcal {K}$.

Proof. By Lemma 2.30, $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$ is closed under limits and colimits in $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$, which are thus computed level-wise, and similarly for $\mathcal {D}_0$. Therefore, we get

\[ f^*(\operatorname*{colim}_I F_i)(c) = (\operatorname*{colim}_I F_i)(fc) \cong \operatorname*{colim}_I F_i(fc) = \operatorname*{colim}_I f^*F_i(c) \cong (\operatorname*{colim}_I f^*F_i)(c) , \]

showing that $f^*$ commutes with colimits, and similarly for limits.

Proof. Lemma 2.33 shows that $f_!\colon \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0) \to \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {D}_0)$ is internally left adjoint.

Proof. Since $\operatorname {Fun}^{\mathcal {K}^\mathrm {op}}((-)^\mathrm {op}, \mathcal {M})\colon \mathrm {Cat}_\mathcal {K}^\mathrm {op} \to \mathrm {Pr}^\mathrm {R}$ is a subfunctor of $\operatorname {Fun}((-)^\mathrm {op}, \mathcal {M})\colon \mathrm {Cat}_\mathcal {K}^\mathrm {op} \to \mathrm {Pr}^\mathrm {R}$, there is a natural transformation from the former to the latter given by the inclusion. Applying [Reference LurieLur17, Corollary 4.7.4.18 (3)] for $S = \mathrm {Cat}_\mathcal {K}^\mathrm {op}$ (and considering our functors as landing in $\widehat {\mathrm {Cat}}$) shows that by passing to the left adjoints in the target and in the natural transformation, we obtain a natural transformation $L_\mathcal {K}\colon \operatorname {\operatorname {\mathcal {P}}^\mathcal {M}} \Rightarrow \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}$. Indeed, $L_\mathcal {K}$ was defined in Definition 2.31 as the left adjoint of the inclusion.

Proof. The natural transformation is obtained by the following diagram.

Here ${\unicode{x3088}}\colon \iota \Rightarrow \operatorname {\mathcal {P}}$ is the ordinary Yoneda natural transformation constructed in [Reference Haugseng, Hebestreit, Linskens and NuitenHHLN20, Theorem 8.1], the natural transformation $u\colon \mathrm {id} \Rightarrow - \otimes \mathcal {M}$ is the unit map of the free-forgetful adjunction $- \otimes \mathcal {M}\colon \mathrm {Pr}^\mathrm {L} \rightleftarrows \mathrm {Mod}_\mathcal {M}\colon \underline {({-})}$ given by tensoring with $\mathcal {S} \to \mathcal {M}$, and $L_\mathcal {K}\colon \operatorname {\operatorname {\mathcal {P}}^\mathcal {M}} \Rightarrow \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}$ is the natural transformation of Proposition 2.36.

Proof. We first reduce to the case where $\mathcal {K} = \emptyset$. Since $L_\mathcal {K}\colon \operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0) \to \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$ is the left adjoint of the inclusion, using Lemma 2.2 we get

\[ \hom^\mathcal{M}({\unicode{x3088}}_\mathcal{K}^\mathcal{M}(X), F) \cong \hom^\mathcal{M}(L_\mathcal{K} {\unicode{x3088}}^\mathcal{M}(X), F) \cong \hom^\mathcal{M}({\unicode{x3088}}^\mathcal{M}(X), F) . \]

We finish the proof by showing that the latter is equivalent to $F(X)$, using the Yoneda lemma in the category $\mathcal {M}$. Indeed, let $m \in \mathcal {M}$ be any object, then

\begin{align*} \hom(m, \hom^\mathcal{M}({\unicode{x3088}}^\mathcal{M}(X), F)) &\cong \hom(m \otimes {\unicode{x3088}}^\mathcal{M}(X), F)\\ &\cong \hom({\unicode{x3088}}^\mathcal{M}(X), \hom^\mathcal{M}(m, F))\\ &\cong \hom({\unicode{x3088}}(X), \hom(m, F))\\ &\cong \hom(m, F)(X)\\ &\cong \hom(m, F(X)) , \end{align*}

where the first and second step use the exponential adjunction, the third uses the free-forgetful adjunction $\mathcal {C} \to \mathcal {C} \otimes \mathcal {M}$, the fourth uses the ordinary Yoneda lemma for $\mathcal {C}_0$ and the last step uses that the action of $\mathcal {M}$ is level-wise.

Proof. By Lemma 2.30, $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$ is closed under colimits and limits in $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$, which are therefore computed level-wise, so that $\hom ^\mathcal {M}({\unicode{x3088}}_\mathcal {K}^\mathcal {M}(X), F) \cong F(X)$ commutes with all colimits in the $F$-coordinate.

Proof. We first show the result for $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$, i.e. for the case $\mathcal {K}=\emptyset$. Recall that $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0) \cong \operatorname {\mathcal {P}}(\mathcal {C}_0) \otimes \mathcal {M}$ is generated under colimits from the image of $\operatorname {\mathcal {P}}(\mathcal {C}_0) \times \mathcal {M}$, i.e. from objects of the form $F \otimes m$. Second, $\operatorname {\mathcal {P}}(\mathcal {C}_0)$ is generated under colimits from objects of the form ${\unicode{x3088}}(X)$ for $X \in \mathcal {C}$. Therefore, $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$ is generated under colimits and the action of $\mathcal {M}$ from objects of the form ${\unicode{x3088}}^\mathcal {M}(X)$ for $X \in \mathcal {C}$, which are indeed atomic by Corollary 2.41.

For the general case, recall that $L_\mathcal {K}\colon \operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0) \to \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$ is an internally left adjoint functor so it sends atomic objects to atomic objects by Proposition 2.13, thus ${\unicode{x3088}}_\mathcal {K}^\mathcal {M}(X)$ is atomic for any $X \in \mathcal {C}$. Since it preserves colimits and the action of $\mathcal {M}$, and ${\unicode{x3088}}^\mathcal {M}(X)$ generate $\operatorname {\operatorname {\mathcal {P}}^\mathcal {M}}(\mathcal {C}_0)$ under these operations, their images ${\unicode{x3088}}_\mathcal {K}^\mathcal {M}(X)$ generate the essential image of $L_\mathcal {K}$ under these operations. In addition, $L_\mathcal {K}$ is essentially surjective, so that ${\unicode{x3088}}_\mathcal {K}^\mathcal {M}(X)$ are atomic generators of $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$ as needed.

Proof. To check that the data in the theorem supports an adjunction, it suffices to check that for any $\mathcal {C}_0 \in \mathrm {Cat}_\mathcal {K}$ and $\mathcal {D} \in \mathrm {Mod}_\mathcal {M}^{\mathrm {iL}}$, the canonical map

(1)\begin{equation} \operatorname{Fun}^\mathrm{iL}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}^\mathcal{M}}(\mathcal{C}_0), \mathcal{D}) \to \operatorname{Fun}_\mathcal{K}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}^\mathcal{M}}(\mathcal{C}_0)^{\mathrm{at}}, \mathcal{D}^{\mathrm{at}}) \xrightarrow{- \circ {\unicode{x3088}}_\mathcal{K}^\mathcal{M}} \operatorname{Fun}_\mathcal{K}(\mathcal{C}_0, \mathcal{D}^{\mathrm{at}}) \end{equation}

is an equivalence (in fact, it suffices to show this for the hom spaces, rather than the functor categories, but we show that the stronger statement holds). Note that

(2)\begin{equation} \operatorname{Fun}^\mathrm{L}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}^\mathcal{M}}(\mathcal{C}_0), \mathcal{D}) \cong \operatorname{Fun}^\mathrm{L}(\operatorname{\operatorname{\mathcal{P}}_\mathcal{K}}(\mathcal{C}_0), \mathcal{D}) \cong \operatorname{Fun}_\mathcal{K}(\mathcal{C}_0, \mathcal{D}) . \end{equation}

Furthermore, both the first and last categories in (1) are full subcategories of the first and last categories in (2), showing that the composition in (1) is also fully faithful.

To finish the argument, we need to show that (1) is essentially surjective. To that end, let $F\colon \mathcal {C}_0 \to \mathcal {D}^{\mathrm {at}}$ be a functor preserving $I^\mathrm {op}$-shaped colimits for $I^\mathrm {op} \in \mathcal {K}$. We can post-compose it with the inclusion $\mathcal {D}^{\mathrm {at}} \to \mathcal {D}$, and using (2) we get a left adjoint functor $\tilde {F}\colon \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0) \to \mathcal {D}$, and we need to show that it is in fact internally left adjoint. By construction, for any $X \in \mathcal {C}_0$ we have that $\tilde {F}({\unicode{x3088}}_\mathcal {K}^\mathcal {M}(X)) \cong F(X) \in \mathcal {D}^{\mathrm {at}}$ is atomic. Proposition 2.42 shows that these are atomic generators for $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}(\mathcal {C}_0)$, so Proposition 2.14 shows that $\tilde {F}$ is indeed internally left adjoint.

Proof. Since $\mathrm {Mod}_\mathcal {M}^{\mathrm {iL}}$ is a wide subcategory of $\mathrm {Mod}_\mathcal {M}$, all we need to show is that if $L_i\colon \mathcal {C}_i \to \mathcal {D}_i$, $i=1,2$, are in $\mathrm {Mod}_\mathcal {M}^{\mathrm {iL}}$, then so is $L_1 \otimes L_2\colon \mathcal {C}_1 \otimes \mathcal {C}_2 \to \mathcal {D}_1 \otimes \mathcal {D}_2$. Let $R_i$ be the right adjoints of $L_i$, which by assumption are themselves left adjoints. Because they are left adjoints, we can tensor them to obtain another left adjoint functor $R_1 \otimes R_2\colon \mathcal {D}_1 \otimes \mathcal {D}_2 \to \mathcal {C}_1 \otimes \mathcal {C}_2$. It is then straightforward to check that tensoring the unit and counit of $L_i \dashv R_i$ exhibit an adjunction $L_1 \otimes L_2 \dashv R_1 \otimes R_2$, showing that $L_1 \otimes L_2$ is an internally left adjoint functor.

Proof. Since the symmetric monoidal structure on $\mathrm {Mod}_\mathcal {M}^{\mathrm {iL}}$ is inherited from $\mathrm {Mod}_\mathcal {M}$, it suffices to show that $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}\colon \mathrm {Cat}_\mathcal {K} \to \mathrm {Mod}_\mathcal {M}$ is symmetric monoidal. Indeed, [Reference LurieLur17, Remark 4.8.1.8] shows that $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}}\colon \mathrm {Cat}_\mathcal {K} \to \mathrm {Pr}^\mathrm {L}$ is symmetric monoidal, $- \otimes \mathcal {M}\colon \mathrm {Pr}^\mathrm {L} \to \mathrm {Mod}_\mathcal {M}$ is symmetric monoidal, and by Proposition 2.29, $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}} \otimes \mathcal {M} \cong \operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}$.

Proof. First, by [Reference Carmeli, Schlank and YanovskiCSY21a, Lemma 5.2.1], the functor $L \otimes \mathrm {id}_\mathcal {N}$ is a localization as well. By [Reference Carmeli, Schlank and YanovskiCSY21a, Proposition 5.2.15], a localization of modes is smashing if and only if it is an internally left adjoint functor in $\mathrm {Pr}^\mathrm {L}$. Thus, $L$ is internally left adjoint, and by Proposition 2.44 we conclude that $L \otimes \mathrm {id}_\mathcal {N}$ is also internally left adjoint, so that it is also a smashing localization.

Proof. Let $\mathcal {K}$ denote the collection of all (small) categories, then by [Reference LurieLur17, Remark 4.8.5.17] there is a symmetric monoidal functor $\mathrm {LMod}_{(-)}\colon \operatorname {Alg}(\mathcal {M}) \to \mathrm {Mod}_\mathcal {M}(\widehat {\mathrm {Cat}}_\mathcal {K})$. As in [Reference LurieLur17, Notation 4.8.5.10], $\mathrm {LMod}_R$ is presentable and $f_!$ is left adjoint to $f^*$. Furthermore, [Reference LurieLur17, Corollary 4.2.3.7 (2)] shows that the right adjoint $f^*$ of $f_!$ is itself a left adjoint, so that the functor lands in $\mathrm {Mod}_\mathcal {M}^\mathrm {iL}$. As the symmetric monoidal structure on $\mathrm {Mod}_\mathcal {M}^\mathrm {iL}$ is restricted from $\mathrm {Mod}_\mathcal {M}(\widehat {\mathrm {Cat}}_\mathcal {K})$, the factorization $\mathrm {LMod}_{(-)}\colon \operatorname {Alg}(\mathcal {M}) \to \mathrm {Mod}_\mathcal {M}^\mathrm {iL}$ is indeed symmetric monoidal.

Proof. We explain how this follows from [Reference LurieLur17, Proposition 4.6.2.18], with $\mathcal {C} = \mathcal {M}$, $A = R^{\mathrm {rev}}$ and the roles of $X$ and $Y$ reversed (see also [Reference LurieLur17, Remark 4.6.3.16]).

For the first direction, assume that there is an adjunction and let $\eta \colon \mathrm {id}_\mathcal {M} \Rightarrow Y \otimes _R X \otimes _{\mathbf {1}_\mathcal {M}} -$ be the unit. By the adjunction, we know that for each $P \in \mathcal {M}$ and $Q \in \mathrm {LMod}_R$ the composition

\[ \hom(X \otimes_{\mathbf{1}_\mathcal{M}} P, Q) \to \hom(Y \otimes_R X \otimes_{\mathbf{1}_\mathcal{M}} P, Y \otimes_R Q) \xrightarrow{- \circ \eta_P} \hom(P, Y \otimes_R Q) \]

is an equivalence. Since both functors in the adjunction preserve colimits, and the categories are in the mode $\mathcal {M}$, the adjunction is $\mathcal {M}$-linear. Therefore, the two maps

\[ \eta_P\colon P \to Y \otimes_R X \otimes_{\mathbf{1}_\mathcal{M}} P ,\quad \eta_{\mathbf{1}_\mathcal{M}} \otimes_{\mathbf{1}_\mathcal{M}} \mathrm{id}_P \colon P \to Y \otimes_R X \otimes_{\mathbf{1}_\mathcal{M}} P \]

coincide. This shows that $c = \eta _{\mathbf {1}_\mathcal {M}}$ satisfies condition $(*)$ of the cited proposition.

Similarly, for the other direction, if $Y$ is left dual to $X$ then the coevaluation map $c\colon \mathbf {1}_\mathcal {M} \to Y \otimes _R X$ gives an ($\mathcal {M}$-linear) natural transformation $c \otimes _{\mathbf {1}_\mathcal {M}} -\colon \mathrm {id}_\mathcal {M} \Rightarrow Y \otimes _R X \otimes _{\mathbf {1}_\mathcal {M}} -$, which is a unit of an adjunction by condition $(*)$ of the cited proposition.

Proof. Consider [Reference LurieLur17, Corollary 4.2.3.7 (2)] where both $\mathcal {C}$ and $\mathcal {M}$ in the reference's notation are our $\mathcal {M}$, $A = \mathbf {1}_\mathcal {C}$ and $B = R$. Then, the functor $\mathrm {LMod}_B \to \mathrm {LMod}_A$ is $\hom ^\mathcal {M}(R, -)\colon \mathrm {LMod}_R \to \mathcal {M}$, which therefore commutes with all colimits, showing that $R$ is atomic. The second part follows from Proposition 2.15.

Proof. The previous lemma shows that $R$ is indeed atomic, and we need to show that it generates $\mathrm {LMod}_R$ under colimits and the action of $\mathcal {M}$. Specifically, we will show that $\mathrm {LMod}_R$ is generated under colimits from $R \otimes m$ for $m \in \mathcal {M}$. By [Reference YanovskiYan22, Corollary 2.5], this is equivalent to showing that $\hom (R \otimes m, -)\colon \mathrm {LMod}_R \to \mathcal {S}$ are jointly conservative. Note that $R \otimes -\colon \mathcal {M} \to \mathrm {LMod}_R$ is the left adjoint of $\underline {({-})}\colon \mathrm {LMod}_R \to \mathcal {M}$, so that $\hom (R \otimes m, -) \cong \hom (m, \underline {({-})})$. These functors are indeed jointly conservative since $\underline {({-})}\colon \mathrm {LMod}_R \to \mathcal {M}$ is conservative, and $\hom (m, -)\colon \mathcal {M} \to \mathcal {S}$ over all $m \in \mathcal {M}$ are jointly conservative.

Proof. Recall that for $X \in \mathrm {LMod}_R$, the functor $X \otimes _{\mathbf {1}_\mathcal {M}} -\colon \mathcal {M} \to \mathrm {LMod}_R$ is left adjoint to $\hom ^\mathcal {M}(X, -)$.

If $X$ is left dualizable, then $X \otimes _{\mathbf {1}_\mathcal {M}} -$ is left adjoint to $Y \otimes _R -$ for some $Y \in \mathrm {RMod}_R$ by Proposition 2.51. By the uniqueness of adjoints we get that $\hom ^\mathcal {M}(X, -) \cong Y \otimes _R -$. Since $Y \otimes _R -$ commutes with colimits, we get that $X$ is atomic.

Now assume that $X$ is atomic. The two functors $\hom ^\mathcal {M}(X, -)$ and $X^\vee \otimes _R -$ are colimit preserving, i.e. morphisms in $\mathrm {Mod}_\mathcal {M}$, thus also commute with tensor from $\mathcal {M}$. Proposition 2.53 shows that $\mathrm {LMod}_R$ is generated from $R$ by these operations, and by the construction of $X^\vee$, they agree on $R$, so the canonical map between the two is an equivalence. This shows that $X \otimes _{\mathbf {1}_\mathcal {M}} -$ is left adjoint to $\hom ^\mathcal {M}(X, -) \cong X^\vee \otimes _R -$, concluding by Proposition 2.51.

Proof. By Proposition 2.54, the $\mathcal {M}$-atomic objects in $\mathcal {M}$ are the dualizable objects. Since $F$ is symmetric monoidal it sends dualizable objects to dualizable objects. Thus, the $\mathcal {M}$-atomic objects are sent to dualizable objects in $\mathcal {N}$. Again by Proposition 2.54, the dualizable objects in $\mathcal {N}$ are $\mathcal {N}$-atomic.

Proof. Follows immediately from the $m$-Segal condition.

Proof. First, the case $\mathcal {C} = \mathcal {S}$ is [Reference LurieLur09, Proposition 5.3.3.3]. Second, the case $\mathcal {C} = \operatorname {\mathcal {P}}(\mathcal {C}_0)$ follows from the previous case, since limits and colimits are computed level-wise in functor categories. Lastly, for the general case we have that $\mathcal {C} \cong \mathrm {Ind}_\kappa (\mathcal {C}^\kappa )$. By [Reference LurieLur09, Proposition 5.3.5.3], $\mathcal {C} \subseteq \operatorname {\mathcal {P}}(\mathcal {C}^\kappa )$ is closed under $\kappa$-filtered colimits. In addition, [Reference LurieLur09, Corollary 5.3.5.4 (3)] shows that it is also closed under limits, since limits commute with limits. To conclude, $\mathcal {C}$ is closed under $\mu$-filtered colimits and $\mu$-small limits in $\operatorname {\mathcal {P}}(\mathcal {C}^\kappa )$ for any $\mu \geq \kappa$, and by the second case the result holds for $\operatorname {\mathcal {P}}(\mathcal {C}^\kappa )$ for any $\mu$.

Proof. We essentially repeat the proof in [Reference HarpazHar20, Lemma 5.17]. Recall that $\mathrm {CMon}_m^{(p)}{({\mathcal {C}})}$ is the full subcategory of $\mathrm {PCMon}_m^{(p)} {({\mathcal {C}})}$ on those functors that preserve limits index by $A \in \mathcal {S}_m^{(p)}$. As limits commute with limits, and limits are computed level-wise in $\mathrm {PCMon}_m^{(p)} {({\mathcal {C}})}$, we get that $\mathrm {CMon}_m^{(p)}{({\mathcal {C}})}$ is closed under limits. Let $\kappa$ be any cardinal such that $\mathcal {C}$ is $\kappa$-presentable and all $A \in \mathcal {S}_m^{(p)}$ are $\kappa$-small, then by Lemma 4.5, limits indexed by $A \in \mathcal {S}_m^{(p)}$ commute with $\kappa$-filtered colimits in $\mathcal {C}$. Again, as colimits are computed level-wise in $\mathrm {PCMon}_m^{(p)} {({\mathcal {C}})}$, we get that $\mathrm {CMon}_m^{(p)}{({\mathcal {C}})}$ is closed under $\kappa$-filtered colimits. It follows that $\mathrm {CMon}_m^{(p)}{({\mathcal {C}})}$ is presentable by the reflection principle of [Reference Ragimov and SchlankRS21]. Alternatively, the presentability follows from [Reference Chu and HaugsengCH21, Lemma 2.11(iv)].

Proof. Follows immediately from the characterization given in Remark 4.3 and the fact that $F$ commutes with these limits.

Proof. The first part is [Reference LurieLur09, Corollary 5.3.6.10] for $\mathcal {K} = \emptyset$ and $\mathcal {K}' = \mathcal {S}_m^{(p)}$. The second part is [Reference LurieLur17, Theorem 1.1.4.4]. The third part follows from the combination of the first two.

Proof. We first show that for any $\mathcal {C} \in \mathrm {Mod}_{\mathcal {M}}$, $\mathcal {C}^{\mathrm {at}}$ is stable and $m$-semiadditive. Recall from [Reference Hopkins and LurieHL13, Proposition 4.3.9] that for any $\mathcal {C}$ in the mode $\mathrm {CMon}_m^{(p)}{({\mathcal {S}})}$ and $m$-finite $p$-space $A$, $A$-shaped limits in $\mathcal {C}$ commute with all colimits, showing that all $m$-finite $p$-space $A$ are absolute limits in $\mathrm {CMon}_m^{(p)}{({\mathcal {S}})}$, and by Lemma 2.20 also in $\mathcal {M}$. Since $\mathcal {C}^{\mathrm {at}}$ is a full subcategory of the $m$-semiadditive category $\mathcal {C}$, and is closed under all colimits indexed by any $m$-finite $p$-space $A$, it is, in fact, $m$-semiadditive by [Reference Carmeli, Schlank and YanovskiCSY21a, Proposition 2.1.4(4)]. The stability statement follows from Proposition 2.25.

In particular, this shows that $\mathcal {K}$, the collection of all finite categories and $m$-finite $p$-spaces, is a collection of absolute limits of $\mathcal {M}$. Theorem 2.46 then shows that there is a lax symmetric monoidal functor $(-)^{\mathrm {at}} \colon \mathrm {Mod}_{\mathcal {M}}^{\mathrm {iL}} \to \mathrm {Cat}_\mathcal {K}$. Recall that there is a fully faithful functor $(-)^\otimes \colon \mathrm {CMon}(\mathrm {Cat})^\mathrm {lax} \to \mathrm {Op}$ from the category of symmetric monoidal categories and lax symmetric monoidal functors to operads. Note that $\mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}} \subset \mathrm {Cat}_\mathcal {K}$ is the full subcategory on those categories which are in addition stable, but it is not a sub-symmetric monoidal category, since the unit of $\mathrm {Cat}_\mathcal {K}$ is not stable. However, it is true that the tensor product of a family of categories in either category is the same, in particular $\mathrm {Cat}_{m\text {-}\mathrm {fin}}^{\mathrm {st},\otimes }$ is a sub-operad of $\mathrm {Cat}_\mathcal {K}^\otimes$. Therefore, we get that the map of operads $(-)^{\mathrm {at}}\colon \mathrm {Mod}_{\mathcal {M}}^{\mathrm {iL},\otimes } \to \mathrm {Cat}_\mathcal {K}^\otimes$ factors through the operad $\mathrm {Cat}_{m\text {-}\mathrm {fin}}^{\mathrm {st},\otimes }$, which corresponds to the desired lax symmetric monoidal functor $(-)^{\mathrm {at}} \colon \mathrm {Mod}_{\mathcal {M}}^{\mathrm {iL}} \to \mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}}$.

Note that ${{\unicode{x05E6}}} ^{[m]} \cong \mathcal {M} \otimes \mathrm {Sp}_{(p)}$ is a smashing localization of $\mathcal {M}$. The argument above works for ${{\unicode{x05E6}}} ^{[m]}$ in place of $\mathcal {M}$. Furthermore, by Proposition 2.26, the atomic objects with respect to either one are the same, showing that $(-)^{\mathrm {at}} \colon \mathrm {Mod}_{{{\unicode{x05E6}}} ^{[m]}}^{\mathrm {iL}} \to \mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}}$ is indeed the restriction of $(-)^{\mathrm {at}} \colon \mathrm {Mod}_{\mathcal {M}}^{\mathrm {iL}} \to \mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}}$. We also note that the lax symmetric monoidal structure on the latter restricts to the lax symmetric monoidal structure on the former. To see that, observe that the symmetric monoidal left adjoint of the latter $\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}}^{{{\unicode{x05E6}}} ^{[m]}}\colon \mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}} \to \mathrm {Mod}_{{{\unicode{x05E6}}} ^{[m]}}^{\mathrm {iL}}$ factors as the composition of the symmetric monoidal functors $\mathrm {Cat}^{\mathrm {st}}_{m\text {-}\mathrm {fin}} \xrightarrow {\operatorname {\operatorname {\mathcal {P}}_\mathcal {K}^\mathcal {M}}} \mathrm {Mod}_{\mathcal {M}}^{\mathrm {iL}} \xrightarrow {- \otimes {{\unicode{x05E6}}} ^{[m]}} \mathrm {Mod}_{{{\unicode{x05E6}}} ^{[m]}}^{\mathrm {iL}}$, so the lax symmetric monoidal right adjoint factors accordingly.

Proof. Fix a map $X \to Y \in \mathcal {D}$. By the Yoneda lemma, $\hom ^\mathcal {D}(Y, T) \to \hom ^\mathcal {D}(X, T)$ is an equivalence for any $T \in \mathcal {D}_0$, if and only if the map $\hom (Z, \hom ^\mathcal {D}(Y, T)) \to \hom (Z, \hom ^\mathcal {D}(X, T))$ is an equivalence for any $Z \in \mathcal {D}, T \in \mathcal {D}_0$. By adjunction, the latter holds if and only if $\hom (Y \otimes Z, T) \to \hom (X \otimes Z, T)$ is an equivalence for any $Z \in \mathcal {D}, T \in \mathcal {D}_0$. By the Yoneda lemma, this holds if and only if $X \otimes Z \to Y \otimes Z$ is an $L$-equivalence for any $Z \in \mathcal {D}$.

Proof. We let $L' = L \otimes \mathrm {id}$. First note that $L'$ is indeed a reflective localization by [Reference Carmeli, Schlank and YanovskiCSY21a, Lemma 5.2.1]. Using [Reference LurieLur17, Proposition 2.2.1.9] we endow $\mathcal {D}_0$ with the localized symmetric monoidal structure, making $L$ into a symmetric monoidal functor. Since $\otimes$ is the coproduct of $\operatorname {CAlg}(\mathrm {Pr}^\mathrm {L})$, this makes the categories and the map $L'\colon \mathcal {D} \otimes \mathcal {E} \to \mathcal {D}_0 \otimes \mathcal {E}$ symmetric monoidal. Now let $X \to Y \in \mathcal {D} \otimes \mathcal {E}$ be an $L'$-equivalence. For any $Z \in \mathcal {D} \otimes \mathcal {E}$, we have