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Proposition 4.8.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces a bijection of sets
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]
- $(2)$
For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces an isomorphism of $\infty $-categories
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]
Proof.
Assume that $(1)$ is satisfied; we will prove $(2)$ (the reverse implication follows immediately from the definitions). Let $\operatorname{\mathcal{D}}$ be an $(n,1)$-category; we wish to show that precomposition with $F$ induces an isomorphism of simplicial sets from $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$ to $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Equivalently, we wish to show that for every simplicial set $K$, the induced map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) ). \]
This follows by applying condition $(1)$ to the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$, which is an $(n,1)$-category by virtue of Proposition 4.8.1.22.
$\square$