Kerodon

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Corollary 8.1.5.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. Then Construction 8.1.5.2 induces an isomorphism of simplicial sets

$\operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y).$

Proof. By definition, a morphism from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})}(X,Y)$ can be identified with a morphism of simplicial sets

$U: \operatorname{N}_{\bullet }([1] \times \operatorname{\mathcal{C}}) \simeq \Delta ^{1} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$

such that $U|_{ \operatorname{N}_{\bullet }( \{ 0\} \times \operatorname{\mathcal{C}})}$ and $U|_{ \operatorname{N}_{\bullet }( \{ 1\} \times \operatorname{\mathcal{D}}) }$ are the constant maps taking the values $X$ and $Y$, respectively. The desired result now follows by combining Theorems 2.3.5.13 and 2.3.4.1. $\square$