Proof. The $\mathcal {V}$ -category of arrows $\mathcal {V}^\to $ is locally $\lambda $ -presentable because $\mathcal {V}$ is. Moreover $\mathcal {M}$ is closed in $\mathcal {V}^\to $ under all conical limits and powers (since $(\mathcal {E},\mathcal {M})$ is enriched); thus is closed under all weighted limits. By [Reference Adámek and Rosický1, Theorem 2.48] $J_0$ has a left adjoint, and, since J preserves powers, the ordinary left adjoint is actually an enriched one. Thus $\mathcal {M}$ is locally $\lambda $ -presentable. It is easy to see that the left adjoint acts as described in the statement.

Assumption 3.1. We fix an enriched factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ whose right class $\mathcal {M}$ is made of monomorphisms and is closed in $\mathcal {V}^\to $ under $\lambda $ -filtered colimits.

Proof. $(1)$ By construction, an object of $\operatorname {\mathbf {Str}}({{\mathbb R}})$ is a $\mathcal {V}$ -functor $\Theta _{{\mathbb R}}\to \mathcal {V}$ whose restriction along $\theta _{{\mathbb R}}$ is isomorphic to $A^{(H_{{\mathbb R}}-)}$ for some $A\in \mathcal {V}$ , and whose restriction along $H_{{\mathbb R}}$ lands pointwise in $\mathcal {M}$ . Now, by definition of $\theta _{\mathbb R}$ , to give the data above is equivalent to give an object $A\in \mathcal {V}$ together with an ordinary functor $2\times {\mathbb R}\to \mathcal {V}_0$ which sends $(1,R:X)$ to $A^X$ , for any $(R:X)\in {\mathbb R}$ , and sends every arrow in $2\times {\mathbb R}$ to a map in $\mathcal {M}$ . In other words, it is equivalent to give an object $A\in \mathcal {V}$ together with an assignment ${\mathbb R}\to \mathcal {M}$ sending a relation $R:X$ to a map in $\mathcal {M}$ with codomain $A^X$ . This is exactly the data of an ${\mathbb R}$ -structure. The same argument applies to morphisms of the underlying category.

$(2)$ The pullback defining $\operatorname {\mathbf {Str}}({{\mathbb R}})$ is a bipullback since $([\theta _{\mathbb R},\mathcal {V}],[H_{\mathbb R},\mathcal {V}]')$ is an isofibration. Moreover, the $\mathcal {V}$ -functors $([\theta _{\mathbb R},\mathcal {V}],[H_{\mathbb R},\mathcal {V}]')$ and $N\times {\prod }_{\mathbb R} J$ are both continuous and $\lambda $ -filtered colimit preserving (by the lemma above), and the $\mathcal {V}$ -categories involved are locally $\lambda $ -presentable (for $\mathcal {M}$ this follows from Lemma 2.1, then use stability of locally $\lambda $ -presentable $\mathcal {V}$ -categories under products [Reference Bird3]). Thus $\operatorname {\mathbf {Str}}({{\mathbb R}})$ is locally $\lambda $ -presentable (again by [Reference Bird3]), and both $U_{\mathbb R}$ and $V_{\mathbb R}$ preserve all limits and $\lambda $ -filtered colimits.

$(3)$ We have already shown that $U_{\mathbb R}$ is continuous and preserves $\lambda $ -filtered colimits. Since $\mathcal {V}\times {\prod }_{\mathbb R}\mathcal {M}$ is locally presentable, then it also has a left adjoint. The fact that $U_{\mathbb R}$ is conservative follows from $(1)$ and the definition of ${\mathbb R}$ -structure.

Proof. This is obtained simply by combining [Reference Rosický and Tendas23, Proposition 3.5] and 3.7 and using the stability of locally $\lambda $ -presentable categories under bilimits [Reference Bird3] (since $U_{\mathbb F}$ and $U_{\mathbb R}$ are isofibrations, the pullback above is also a bipullback).

Proof. It is easy to see that F and R define respectively left and right adjoints to the underlying ordinary functor of $J_{\mathbb F}$ , and that they are fully faithful as ordinary functors. To conclude, by [Reference Kelly10, Theorem 4.85], it is enough to prove that $J_{\mathbb F}$ preserves powers and copowers by elements of $\mathcal {V}$ . The former follows by definition since $J_{\mathbb F}$ is obtained by pulling back continuous functors. For the latter, consider an ${\mathbb L}$ -structure A and some $Y\in \mathcal {V}$ ; then the copower $X\cdot A$ of A by X can be constructed by considering the copower $X\cdot J_{\mathbb FA}$ in $\operatorname {\mathbf {Str}}({{\mathbb F}})$ , and by extending it to an ${\mathbb L}$ -structure as follows: for any relation $R:Y$ we define $r_{X\cdot A}\colon R_{X\cdot A}\rightarrowtail (X\cdot A)^Y$ as the $\mathcal {M}$ -subobject obtained by taking the $(\mathcal {E},\mathcal {M})$ -factorization of the composite

$$ \begin{align*}X\cdot R_A \xrightarrow{X\cdot r_A} X\cdot (A^Y)\longrightarrow (X\cdot A)^Y\end{align*} $$

in $\mathcal {V}$ . It follows that $J_{\mathbb F}$ also preserves copowers.

Proof. It is enough to work at the level of the underlying ordinary category $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ since enriched and ordinary finitely presentable objects coincide [Reference Kelly10, Proposition 7.5]. Let $\mathcal {G}$ be the full subcategory of $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ with objects those A satisfying the three properties above. To conclude it is enough to show (1) that every object of $\mathcal {G}$ is $\lambda $ -presentable, (2) that $\mathcal {G}$ is closed under $\lambda $ -small colimits in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ , and (3) that $\mathcal {G}$ is strongly generating.

(1). Let $A\in \mathcal {G}$ and $h\colon A\to L:=\operatorname {colim}_iL_i$ be a morphism into a filtered colimit of objects $L_i$ in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ , denote the colimiting cone by $(c_i\colon L_i\to L)_i$ . Since $J_{\mathbb {F}}(A)$ is $\lambda $ -presentable and $J_{\mathbb {F}}$ preserves $\lambda $ -filtered colimits, $J_{\mathbb {F}}(h)$ factors through some $J_{\mathbb {F}}(c_i)$ as a morphism $k\colon J_{\mathbb {F}}(A)\to J_{\mathbb {F}}(L_i)$ of ${\mathbb F}$ -structures. We need to extend k to a map of ${\mathbb L}$ -structures, maybe changing index i.

Going back in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ , for every relation $R:X$ we have the solid part of diagram below

with $Y\in \mathcal {V}_\lambda $ . Now, if $Y=0$ , then $k_R$ is the unique map into $R_{L_i}$ induced by $!\colon 0\to R_{L_i}$ and the orthogonality property of the factorization system. Otherwise, since $R_L$ is the $\lambda $ -filtered colimit of the $R_{L_i}$ , the map $h_Rq$ factors through some $(c_j)_R\colon R_{L_j}\to R_L$ . Without loss of generality we can assume $i=j$ and hence we have a map $s\colon Y\to R_{L_i}$ such that $ r_{L_i}s= (k^Xr_A)q$ . Since q is in $\mathcal {E}$ and and $r_{L_i}$ in $\mathcal {M}$ , there is an induced $k_R\colon R_A\to R_{L_i}$ making the square in the diagram above commute. Since there are only a $\lambda $ -small number relations which are not $\mathcal {E}$ -quotients of $0$ , we can repeat this argument and obtain a map $k\colon A\to L_i$ in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ such that $c_ik=h$ . By construction, any two such factorizations $k\colon A\to L_i$ and $k'\colon A\to L_j$ of h must coincide after composition with some maps $i\to l$ and $j\to l$ . This proves that A is $\lambda $ -presentable in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ .

(2). Since $J_{\mathbb F}$ is cocontinuous, the objects of $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ satisfying the first condition are clearly closed under $\lambda $ -small colimits.

In general, given a $\lambda $ -small limit $A:=\operatorname {colim}_i(A_i)$ in $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ and a relation symbol $R:X$ in ${\mathbb L}$ , the $\mathcal {M}$ -subobject $R_A$ of $A^X$ is determined by the $(\mathcal {E},\mathcal {M})$ -factorization of the induced map $\operatorname {colim}_i(R_{A_i})\to A^X$ . It follows that $R_A$ is an $\mathcal {E}$ -quotient of $\operatorname {colim}_i(R_{A_i})$ . Thus, if each $R_{A_i}$ is an $\mathcal {E}$ -quotient of some $Y_i\in \mathcal {V}_\lambda $ , then $\operatorname {colim}_i(R_{A_i})$ is an $\mathcal {E}$ -quotient of $Y:=\sum _iY_i$ , which is still in $\mathcal {V}_\lambda $ . It follows that $R_A$ is also an $\mathcal {E}$ -quotient of Y. Thus the objects satisfying the second condition are clearly closed under $\lambda $ -small colimits.

Since $0$ is stable under $\lambda $ -small colimits, then also are its $\mathcal {E}$ -quotients (by the arguments above); therefore also the objects satisfying the third condition are closed under $\lambda $ -small colimits.

(3). Notice that, since the projections into $\operatorname {\mathbf {Str}}({{\mathbb F}})$ and into $\mathcal {M}$ (for any relation in ${\mathbb L}$ ) are jointly conservative, to show that $\mathcal {G}$ is a strong generator of $\operatorname {\mathbf {Str}}({{\mathbb L}})_0$ it is enough to show the following for any ${\mathbb L}$ -structure L. (a) That any morphism of ${\mathbb F}$ -structures $B\to J_{\mathbb F}(L)$ , with $B \lambda $ -presentable, factors through some $J_{\mathbb F}(A\to L)$ with $A\in \mathcal {G}$ . (b) That, for any relation $R:X$ , any morphism $m\to r_L$ in $\mathcal {M}$ , with $m\in \mathcal {M}_\lambda $ , factors through the R-component of some $A\to L$ with $A\in \mathcal {G}$ .

Point (a) is trivial since, by definition $J_{\mathbb F}(\mathcal {G})= \operatorname {\mathbf {Str}}({{\mathbb F}})_\lambda $ . For (b), consider a relation $R:X$ and any morphism $(u,v)\colon m\to r_L$ in $\mathcal {M}$ , with $m\colon B\rightarrowtail C$ in $\mathcal {M}_\lambda $ ; so that $u\colon B\to R_L$ , $v\colon C\to L^X$ , and $r_Lu=vm$ . Note that $m\colon B\rightarrowtail C$ is $\lambda $ -presentable in $\mathcal {M}$ if and only if $C\in \mathcal {V}_\lambda $ and B is an $\mathcal {E}$ -quotient of some $Y\in \mathcal {V}_\lambda $ .

Since we can write the ${\mathbb F}$ -structure $J_{\mathbb F}(L)$ as a $\lambda $ -filtered colimit of $\lambda $ -presentable ${\mathbb F}$ -structures, and this colimit is computed as in $\mathcal {V}$ , the map v factors through some $h^X\colon A^X\to L^X$ arising from an $h\colon A\to L$ in $\operatorname {\mathbf {Str}}({{\mathbb F}})$ . Therefore we have the solid part of the diagram below

and we define $R_A$ to be the $(\mathcal {E},\mathcal {M})$ -factorization of the composite $B\to A^X$ and the map $h_R\colon R_A\to R_L$ is induced by orthogonality of $\mathcal {E}$ and $\mathcal {M}$ . It follows that we can extend A to an ${\mathbb L}$ -structure in $\mathcal {G}$ by defining the interpretation of $R:X$ as $R_A$ (which is an $\mathcal {E}$ -quotient of $Y\in \mathcal {V}_\lambda $ ) and setting all the other relations to be $\mathcal {E}$ -quotients of $0$ . By construction h extends to a morphism of ${\mathbb L}$ -structures which satisfies our condition.

Notation 4.1. Variables have arities which are objects of $\mathcal {V}$ and we denote them as $x:X$ , for $X\in \mathcal {V}$ ; as extended terms they correspond to the identity maps between arities. Then, given an $(X,Z)$ -ary term t we write $t(x)$ to emphasize that t has input arity X. If t has input arity $X+Y$ we write $t(x,y)$ to specify that we have two input variables; this helps notationally for existential quantification.

Notation 4.7. If s and t are respectively $(X,Z)$ -ary and $(Y,Z)$ -ary terms, then we denote by $(s(x)=t(y))$ the $(X+Y)$ -ary atomic formula defined by

$$ \begin{align*}(s(x)=t(y)):=(\ s(i_X(x,y))=t( i_Y(x,y))\ )\end{align*} $$

where $i_X$ and $i_Y$ are respectively the inclusions of X and Y in $X+Y$ . It is easy to see that the interpretation $\varphi _A\rightarrowtail A^{X+Y}\cong A^X\times A^Y$ is defined by the $(\mathcal {E},\mathcal {M})$ -factorization of the map

induced by the pullback of $s_A$ along $t_A$ (meaning that h is the morphism induced into the product).

Proof. These follow directly from how we defined interpretation starting from atomic formulas, using that $(\mathcal {E},\mathcal {M})$ -factorizations are stable under the limits and colimits considered in the statement (given the additional hypotheses).

Notation 4.17. Let $\psi (x,y)$ be an $X+Y$ -ary formula. We say that an ${\mathbb L}$ -structure A satisfies $(\exists !y)\psi (x,y$ ) if it satisfies the formula $(\exists y)\psi (x,y)$ and the sequent

$$ \begin{align*}(\forall x)(\forall y)(\forall y')(\psi(x,y)\wedge\psi(x,y')\ \vdash\ y=y'). \end{align*} $$

When considering a sequent of the form

$$ \begin{align*}(\forall x)( \varphi(x) \vdash (\exists !y)\psi(x,y)), \end{align*} $$

we say that A satisfies it if it satisfies

$$ \begin{align*}(\forall x)( \varphi(x) \vdash (\exists y)\psi(x,y)), \end{align*} $$

and

$$ \begin{align*}(\forall x)(\forall y)(\forall y')( \varphi(x) \wedge \psi(x,y)\wedge\psi(x,y')\ \vdash \ y=y'). \end{align*} $$

Assumption 5.1. We fix a proper enriched factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which is closed in $\mathcal {V}^\to $ under $\lambda $ -filtered colimits.

Proof. By definition of satisfaction, $\varphi $ is a presentation for A if and only if

$$ \begin{align*}\operatorname{\mathbf{Str}}({{\mathbb L}})_0(A,-)\cong \mathcal {V}_0(I,(\varphi_{(-)})_0).\end{align*} $$

But $\operatorname {\mathbf {Str}}({{\mathbb L}})_0(A,-)$ is by definition $\mathcal {V}_0(I,\operatorname {\mathbf {Str}}({{\mathbb L}})(A,-)_0)$ , and both $\operatorname {\mathbf {Str}}({{\mathbb L}})(A,-)$ and $\varphi _{(-)}$ preserve powers (being continuous); thus the existence of the natural isomorphism above is equivalent to having

$$ \begin{align*}\mathcal {V}_0(X,\operatorname{\mathbf{Str}}({{\mathbb L}})(A,-)_0)\cong \mathcal {V}_0(X,(\varphi_{(-)})_0)\end{align*} $$

naturally in $X\in \mathcal {V}_0$ . And this is in turn equivalent to $\operatorname {\mathbf {Str}}({{\mathbb L}})(A,-)\cong \varphi _{(-)}$ .

Proof. The $\mathcal {V}$ -functor $ \varphi _{(-)}\colon \operatorname {\mathbf {Str}}({{\mathbb L}})\longrightarrow \mathcal {V}$ is continuous and preserves $\lambda $ -filtered colimits since it is defined by taking $\lambda $ -small limits of the forgetful $\mathcal {V}$ -functors U and V. Thus $\varphi _{(-)}$ has a left adjoint, and is therefore a representable $\mathcal {V}$ -functor. It follows that the representing object A is $\lambda $ -presentable and, by the proposition above, $\varphi $ is a presentation formula for A.

Proof. $(1)$ . Consider a morphism $g\colon A\to B$ in $\operatorname {\mathbf {Str}}({\mathbb L})$ , and the $(\mathcal {E},\mathcal {M})$ factorization $(e\colon A\to E, m\colon E\to B)$ of $Ug$ in $\mathcal {V}$ . By [Reference Rosický and Tendas23, Lemma B.5(1)] we know that E inherits a unique notion of ${\mathbb F}$ -structure making e and m morphisms in $\operatorname {\mathbf {Str}}( {\mathbb F})$ ; thus we are left to prove that E also inherits a compatible notion of ${\mathbb R}$ -structure.

For any X-ary relation symbol R in ${\mathbb L}$ , we can consider the solid part of the diagram below in $\mathcal {V}$ .

Then we define $R_E$ , $e_R$ , and $m_R$ as the $(\mathcal {E},\mathcal {M})$ -factorization of $g_R$ , and $r_E$ as the morphism induced by orthogonality. Following [Reference Freyd and Kelly7, Proposition 2.1.4], $r_E$ is in $\mathcal {M}$ . This endows E with an ${\mathbb L}$ -structure that by construction makes e and m morphisms in $\operatorname {\mathbf {Str}}({{\mathbb L}})$ .

$(2)$ . Given a regular epimorphism $g\colon A\to B$ in $\operatorname {\mathbf {Str}}({\mathbb L})$ we can consider the factorization $g=me$ as above. But m is a monomorphism and g is in particular an extremal epimorphism; thus m is an isomorphism and g is therefore sent to a map in $\mathcal {E}$ .

Proof. Let $\mathcal {G}$ be the full subcategory of $\operatorname {\mathbf {Str}}({\mathbb F})$ spanned by these objects. We need to prove that it consists of all the $\lambda $ -presentable objects. Clearly $\mathcal {G}\subseteq \operatorname {\mathbf {Str}}({\mathbb F})_\lambda $ and is a strong generator of $\operatorname {\mathbf {Str}}({\mathbb F})$ , since it contains the free $\lambda $ -presentable objects. Thanks to [Reference Kelly10, Theorem 7.2], to conclude it is enough to prove that $\mathcal {G}$ is closed under $\lambda $ -small weighted colimits.

Since $\lambda $ -small coproducts and copowers of free $\lambda $ -presentable objects are still free, it is easy to see that $\mathcal {G}$ is closed under these colimits. Thus we only need to consider coequalizers. Consider a pair $f,g\colon A\to B$ in $\mathcal {G}$ , then we can consider the solid part of the diagram below

where the pair $(h,k)$ presents B as a coequalizer of free $\lambda $ -presentable objects, while Z is $\lambda $ -presentable $\mathcal {E}$ -projective and r an epimorphism (this exists by hypothesis since $A\in \mathcal {G}$ and the $\mathcal {E}$ -projectives are an $\mathcal {E}$ -generator in $\mathcal {V}_\lambda $ ). Now, by the lemma above, $Uq\in \mathcal {E}$ ; therefore using the $\mathcal {E}$ -projectivity of Z we find $f',g'\colon FZ\to FW$ making the square above commute. To conclude it is enough to notice that the coequalizer of the pair

coincides with the coequalizer of $(f,g)$ .

Proof. Given a $\lambda $ -presentable ${\mathbb L}$ -structure A, we will construct its presentation formula. Since the algebraic reduct $J_{{\mathbb F}}(A)$ is a $\lambda $ -presentable, it is a coequalizer

of morphisms f and g between free algebras over $\lambda $ -presentable objects $Z'$ and Z of $\mathcal {V}$ (see 5.6). Since f and g are $(Z,Z')$ -ary terms, A satisfies the equation $f=g$ .

Now, by 3.13 there are a $\lambda $ -small number of $X_t$ -ary relations $R_t$ such that $(R_t)_A$ is an $\mathcal {E}$ -quotient $q_t:Y_t\twoheadrightarrow (R_t)_A$ of some $0\neq Y_t\in \mathcal {V}_\lambda $ . Since by hypothesis $\mathcal {V}_\lambda $ has an $\mathcal {E}$ -generator of $\mathcal {E}$ -projective objects, we can assume that each $Y_t$ is $\mathcal {E}$ -projective. It follows that for each t we have the following diagram

in $\mathcal {V}$ , where $\tau _t'$ exists since $U(h)^{X_t}$ is in $\mathcal {E}$ (by 5.5 and since $X_t$ is $\mathcal {E}$ -stable) and $Y_t$ is $\mathcal {E}$ -projective. By transposition $\tau _t'$ corresponds to a map $\tau _t"\colon F(X_t\otimes Y_t)\to FZ$ , and hence to a $(Z,X_t\otimes Y_t)$ -ary term $\tau _t$ . Now for each t, consider the Z-ary atomic formula $R_t^{Y_t}(\tau _t)$ . We shall show that the formula

$$ \begin{align*}\pi^A:=(f=g) \wedge \bigwedge\limits_{t} R_t^{Y_t}(\tau_t)\end{align*} $$

is a presentation formula for A. We need to show that for every ${\mathbb L}$ -structure B we have a bijection, natural in B, between elements $a\colon Z\to B$ for which $B\models \pi ^A[a]$ and morphisms $\bar a\colon A\to B$ of ${\mathbb L}$ -structures.

Notice first that $(f=g)_B$ is by definition the equalizer of $B^f,B^g\colon B^Z\to B^{Z'}$ and this, since $B^{(-)}\cong {\mathbb F}\mathrm {-Str}(F-,J_{\mathbb FB})$ , is isomorphic to ${\mathbb F}\mathrm {-Str}(J_{\mathbb FA},J_{\mathbb FB})$ . Thus we have a natural bijection between elements $a\colon Z\to B$ in $\mathcal {V}$ for which $B\models (f=g)[a]$ and morphisms of ${\mathbb F}$ -structures $\bar a\colon J_{\mathbb FA}\to J_{\mathbb FB}$ . To conclude it is enough to show that for such an $a\colon Z\to B$ , we have $B\models R_t^{Y_t}(\tau _t)[a]$ for all t if and only if $\bar a$ is actually a morphism of ${\mathbb L}$ -structures.

Given $a\colon Z\to B$ , denote by $a'\colon I\to B^Z$ its transpose; by naturality, we know that the transpose of the morphism

$$ \begin{align*}I\xrightarrow{\ a'\ } B^Z\xrightarrow{\ B^\tau} B^{X_t\otimes Y_t}\end{align*} $$

with respect to $Y_t$ , is the composite

$$ \begin{align*}x_t\colon Y_t\xrightarrow{\ q_t\ } (R_t)_A\xrightarrow{\ (r_t)_A\ } A^{X_t}\xrightarrow{\ \bar a^{X_t}} B^{X_t}.\end{align*} $$

Now, since $q_t$ is in $\mathcal {E}$ , the map of ${\mathbb F}$ -structure $\bar a$ extends to a map of ${\mathbb L}$ -structures if and only if $x_t\colon Y_t\to B^{X_t}$ above factors through $(r_t)_B\colon (R_t)_B\rightarrowtail B^{X_t}$ for any t (the morphism $(R_t)_A\to (R_t)_B$ is induced by the orthogonality property).

On the other hand, the interpretation of $R_t^{Y_t}(\tau _t)$ in B is given by the pullback

where we have identified $B^{X_t\otimes Y_t}$ with $(B^{X_t})^{Y_t}$ . Thus $a\colon Z\to B$ is such that $B\models R_t^{Y_t}(\tau _t)[a]$ if and only if $B^\tau \circ a'\colon I\to B^{X_t\otimes Y_t}$ factors through $(r_t)_B^{Y_t}$ . By transposing with respect to $Y_t$ , that holds if and only if $x_t\colon Y_t\to B^{X_t}$ factors through $(r_t)_B\colon (R_t)_B\rightarrowtail B^{X_t}$ . By the argument above this holds for any t if and only if $\bar a$ extends to a map of ${\mathbb L}$ -structures, concluding the proof.

Proof. Since the functional part of the language is trivial, by Proposition 3.13, an ${{\mathbb L}}$ -structure A is $\lambda $ -presentable if and only if A is $\lambda $ -presentable as an object of $\mathcal {V}$ and there are a $\lambda $ -small number of $X_t$ -ary relations $R_t$ such that $(R_t)_A$ is an $\mathcal {E}$ -quotient $q_t:Y_t\twoheadrightarrow (R_t)_A$ of some $0\neq Y_t\in \mathcal {V}_\lambda $ .

Thus we a can argue exactly as in Proposition 5.7 above by taking $f=g=1_A$ and $\tau ^{\prime }_t=(r_t)_A q_t$ .

Proof. Assume first that ${{\mathbb L}}$ has non empty functional part (so that we are in the setting of 5.7). Let

and

be the coequalizers from the proof of 5.7. There is an $\mathcal {E}$ -projective object X and a morphism $e:X\to Z_A$ in $\mathcal {E}$ . Then $FX$ is projective with respect to $h_B$ (since $Uh_B$ is in $\mathcal {E}$ by 5.5). Thus, there is $t:FX\to FZ_B$ such that $h_Bt=hh_AF(e)$ . Let $\tau $ be the $(Z_B,X)$ -ary term corresponding to t and $\varepsilon $ be the $(Z_A,X)$ -ary term corresponding to $Fe$ . Then we take $\pi ^A$ and $\pi ^B$ as in 5.7 and define

$$ \begin{align*}\chi'(x,y):=(\pi^A(x)\ \wedge\ \pi^B(y)\ \wedge\ (\tau(y)=\varepsilon(x))\ ).\end{align*} $$

and $\chi (x):= (\exists y)\chi '(x,y)$ ; here, x is a $Z_A$ -sorted variable, y is a $Z_B$ -sorted variable and the equation $\tau (y)=\varepsilon (x)$ is in the sense of 4.7.

When the functional part of ${{\mathbb L}}$ is empty (and we are in the setting of 5.8) we define $\chi $ in the same way, noting that by 5.8 the chosen coequalizers are trivial now, so we do not need to consider an $\mathcal {E}$ -projective object covering $Z_A=A$ (hence $e=1_{A}$ and $t=h$ ).

By construction for each K we have $\operatorname {\mathbf {Str}}({\mathbb L})(A,K)\cong \pi ^A_K$ and $\operatorname {\mathbf {Str}}({\mathbb L})(B,K)\cong \pi ^B_K$ . Consider the solid part of the diagram below, where the squares (I) and (II) are pullbacks by definition, and $\chi "(x,y):= \pi ^B(y) \wedge (\tau (y)=\epsilon (x))$ .

Note that, commutativity of the outer square, plus the fact that (I) is a pullback, implies that the composite $\pi ^h_Km$ factors through $(\tau =\epsilon )_K\to K^{Z_A}$ to give a map $r'\colon \pi ^B_K\to (\tau =\epsilon )_K$ . Similarly, $r'$ factors through $\chi ^{\prime \prime }_K$ first, and then through $\chi ^{\prime }_K$ providing a morphism $r\colon \pi ^B_K\to \chi ^{\prime }_K$ for which $ nqr=m\pi ^h_K $ and $sr=\operatorname {id}$ . Since $e\in \mathcal {E}$ is (in particular) an epimorphism, then $K^e$ is a monomorphism, and hence so is s (being obtained by pulling back $K^e$ and composing with a map in $\mathcal {M}$ ). It follows that s is an isomorphism with inverse given by r. Now, by orthogonality of the factorization system there exists $g\colon \chi _K\to \pi ^A_K$ in $\mathcal {M}$ making the relevant triangles commute. This concludes the proof implying both points (3) and (4).

For the last statement of the lemma in the context of 5.10, one considers the formulas $\pi ^A(x)$ , $\pi ^B(y)$ , and $\chi (x)$ to be those obtained by applying the result above with respect to the factorization system $(\mathcal {E}_0,\mathcal {M}_0)$ . By 5.10 these still satisfy (1) and (2). Then points (3) and (4) can be proved as above.

Assumption 6.1. We fix a proper enriched factorization system $(\mathcal {E}_0,\mathcal {M}_0)$ on $\mathcal {V}$ for which $\mathcal {M}_0$ is closed in $\mathcal {V}^\to $ under $\lambda $ -filtered colimits, and we assume that either of the following two conditions holds:

• • ${{\mathbb L}}={{\mathbb R}}$ is a $\lambda $ -ary relational language;

• • ${\mathbb L}=$ is a $\lambda $ -ary language whose function symbols have $\mathcal {E}_0$ -stable input arities and relation symbols have $\mathcal {E}_0$ -stable arities. Moreover $\mathcal {V}_\lambda $ has an $\mathcal {E}_0$ -generator of $\mathcal {E}_0$ -projective objects.

For the reminder of this section we fix an enriched factorization system $(\mathcal {E},\mathcal {M})$ as in 5.1, such that $\mathcal {E}_0\subseteq \mathcal {E}$ and for which the inclusion $ J \colon \operatorname {\mathbf {Str}}({\mathbb L})\hookrightarrow \operatorname {\mathbf {Str}}({\mathbb L})^0$ preserves $\lambda $ -presentable objects. Structures and formulas will be considered with respect to the factorization system $(\mathcal {E},\mathcal {M})$ .