Kerodon

Remark 4.4.4.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and suppose we are given a pair of functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, which restrict to functors between their cores $F^{\simeq }, G^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ (see Remark 4.4.3.5). Let $u$ be a natural transformation from $F$ to $G$, which we identify with a map of simplicial sets $u: \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. If $u$ is a natural isomorphism, then it restricts to a map of simplicial sets $u_0: \Delta ^1 \times \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$, which we can regard as a homotopy from $F^{\simeq }$ to $G^{\simeq }$. In particular, if the functors $F$ and $G$ are naturally isomorphic, then the morphisms $F^{\simeq }$ and $G^{\simeq }$ are homotopic.