In common parlance, a construction is homotopical if it is invariant under weak equivalence. A generic functor frequently does not have this property. In certain cases, the functor can be approximated by a derived functor, a notion first introduced in homological algebra, which is a universal homotopical approximation either to or from the original functor.

The definition of a total derived functor is simple enough: it is a Kan extension whose handedness unfortunately contradicts that of the derived functor, along the appropriate localization (see Example 1.1.11). But, unusually for constructions characterized by a universal property, generic total derived functors are poorly behaved: for instance, the composite of the total left derived functors of a pair of composable functors is not necessarily a total left derived functor for the composite. The problem with thes tandard definition is that total derived functors are not typically required to be pointwise Kan extensions. In light of Theorem 1.3.5, this seems reasonable, because homotopy categories are seldom complete or cocomplete.

One of the selling points of Daniel Quillen's theory of model categories is that they highlight classes of functors – the left or right Quillen functors – whose left or right derived functors can be constructed in a uniform way, making the passage to total derived functors pseudofunctorial. However, it turns out a full model structure is not necessary for this construction, as suggested by the slogan that “all that matters are the weak equivalences.”