ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 663Here is one example of the usefulness of Theorem 8.8. We say that a V -functorf : A → B is fully faithful if each morphism A (x, y) → B(f(x), f(y)) is cartesian overɛf x ×ɛf y . It is easy to see that this is equivalent to the induced morphism A → B(f, f) ofprofunctors A −↦−→ A being an isomorphism, or equivalently that B(1, f) ⊙ B(f, 1) ∼ = A .8.9. Corollary. Left and right Kan extensions along fully faithful V -functors are honestextensions. In other words, if j : A → K is fully faithful and l: K → C is a left Kanextension of f : A → C along j, then lj ∼ = f.Proof. Left Kan extensions are K(j, 1)-weighted colimits, while precomposition with jis a K(1, j)-weighted colimit. Thus, by Theorem 8.8, lj is a (K(1, j) ⊙ K(j, 1))-weightedcolimit. However, since j is fully faithful, (K(1, j) ⊙ K(j, 1)) is isomorphic to the identityprofunctor, for which a weighted colimit of f is just f itself. The case of right Kanextensions is dual.We now consider what it means for a V -functor to preserve or reflect limits. LetJ : K −→ −↦ A be a weight and d: A → C a V -functor, and suppose given a bimorphismC (l, 1), J ψ −→ C (d, 1) (8.10)Let f : C → D be a V -functor, and consider the unique morphismC (fl, 1), J −→ C (fd, 1) (8.11)whose composite with the universal bimorphism C (f, 1), C (l, 1) φ −→ C (fl, 1) isC (f, 1), C (l, 1), J 1,ψ−→ C (f, 1), C (d, 1) φ −→ C (fd, 1).8.12. Definition. In the above situation, if (8.10) exhibits l as a J-weighted colimit ofd, we say that f preserves this colimit if (8.11) exhibits fl as a J-weighted colimit offd. Similarly, if (8.11) exhibits fl as a J-weighted colimit of fd, we say that f reflectsthis colimit if (8.10) exhibits l as a J-weighted colimit of d.Dually, we define what it means for a V -functor to preserve and reflect a weightedlimit. The following observations are expected.8.13. Proposition. If f : C → D is a left adjoint, then f preserves any colimits whichexist in C . Dually, right adjoints preserve all limits.Proof. Recall that an adjunction f ⊣ g implies an isomorphism D(f, 1) ∼ = C (1, g).Therefore, if l: K → C is a colimit of d: A → C weighted by J : K −→ −↦ A, then for any