Kerodon

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Remark 4.5.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $X$ be a simplicial set. Using Theorem 4.4.4.4, we see that the natural identification $\operatorname{Fun}(X, \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \simeq \operatorname{Fun}( \Delta ^1, \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )$ restricts to an isomorphism $\operatorname{Fun}(X, \operatorname{Isom}(\operatorname{\mathcal{C}}) ) \simeq \operatorname{Isom}( \operatorname{Fun}(X, \operatorname{\mathcal{C}}) )$. If $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ are functors of $\infty$-categories, we obtain a canonical isomorphism

$\operatorname{Fun}(X, \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1) \simeq \operatorname{Fun}(X, \operatorname{\mathcal{C}}_0) \times ^{\mathrm{h}}_{\operatorname{Fun}(X,\operatorname{\mathcal{C}})} \operatorname{Fun}(X, \operatorname{\mathcal{C}}_1).$