# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 7.1.3.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every simplicial set $K$, the functor $F$ preserves $K$-indexed colimits and the functor $G$ preserves $K$-indexed limits.

Proof. We will show that $F$ preserves $K$-indexed colimits; the assertion that $G$ preserves $K$-indexed limits can be proved by a similar argument. Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, so that $F$ induces a functor $F': \operatorname{\mathcal{C}}_{u/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ u) / }$. We wish to show that the functor $F'$ carries initial objects of $\operatorname{\mathcal{C}}_{u/}$ to initial objects of $\operatorname{\mathcal{D}}_{ (F \circ u)/}$. It follows from Corollary 6.2.4.6 that the functor $F'$ also admits a right adjoint. We may therefore replace $F$ by $F'$ and thereby reduce to the case where $K = \emptyset$. In this case, we must show that if $X$ is an initial object of $\operatorname{\mathcal{C}}$, then $F(X)$ is an initial object of $\operatorname{\mathcal{D}}$. Choose an object $Y \in \operatorname{\mathcal{D}}$; we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F( X ), Y )$ is a contractible Kan complex. Proposition 6.2.1.17 supplies a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$. We conclude by observing that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) )$ is contractible, by virtue of our assumption that the object $X \in \operatorname{\mathcal{C}}$ is initial. $\square$