Remark 4.5.1.16. Let $X$ be an arbitrary simplicial set. Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ determines a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ to itself. In particular, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories, then the induced map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})$ is also an equivalence of $\infty $-categories.
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