Remark 8.3.5.9. In the situation of Corollary 8.3.5.8, suppose that we are given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. The commutativity of (8.55) guarantees that the diagram
commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the vertical maps are the isomorphisms of Remark 8.3.5.4. 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 $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$).