Corollary 8.1.8.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. Then Construction 8.1.8.2 induces an isomorphism of simplicial sets
\[ \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y). \]