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

Remark In the situation of Corollary, suppose that we are given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. The commutativity of (8.58) guarantees that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \ar [r]^-{F} \ar [d]^{ \sim } & \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [d]^{\sim } \\ \mathscr {H}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{ \gamma } & \mathscr {H}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) } \]

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the vertical maps are the isomorphisms of Remark We can summarize the situation more informally as follows: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor between (locally small) $\infty $-categories, then the induced map of Kan complexes $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) )$ depends functorially on the pair $(X,Y)$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$).