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Corollary 4.3.5.17. Let $\operatorname{\mathcal{C}}$ be a category and let $K$ be a simplicial set equipped with a morphism $f: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $u: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{K}}$ as a homotopy category of $K$ (see Definition 1.2.5.1), so that $f$ factors uniquely as a composition $K \xrightarrow {u} \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow {\operatorname{N}_{\bullet }(F)} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ for some functor $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$. Then $u$ induces isomorphisms of simplicial sets

\[ \theta : \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/\operatorname{N}_{\bullet }(F)} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f} \quad \quad \theta ': \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{F/} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{\operatorname{N}_{\bullet }(F)/} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{f/}. \]

Proof. We will prove that $\theta $ is an isomorphism; the proof for $\theta '$ is similar. Fix an $n$-simplex $\sigma $ of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f}$, which we identify with a morphism of simplicial sets $\overline{f}: \Delta ^ n \star K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ satisfying $\overline{f}|_{K} = f$. Let $\overline{f}_0 = \overline{f}|_{ \Delta ^ n }$. Using Proposition 4.3.5.13, we can identify $\overline{f}$ with a morphism of simplicial sets $g: K \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{\overline{f}_0 / }$. We wish to show that $\sigma $ can be lifted uniquely to an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ / \operatorname{N}_{\bullet }(F) }$. Equivalently, we wish to show that $g$ admits a unique factorization

\[ K \xrightarrow {u} \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow { \overline{g} } \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{ \overline{f}_0 / } \]

for which the composite map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow { \overline{g} } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ \overline{f}_0 / } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is equal to $\operatorname{N}_{\bullet }(F)$. This follows our assumption that $u$ exhibits $\operatorname{\mathcal{K}}$ as a homotopy category of $K$, since the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ \overline{f}_0 / }$ is isomorphic to the nerve of a category (see Example 4.3.5.7). $\square$