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Exercise 4.6.4.15. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $f$ can be identified with a vertex of the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$, which (to avoid confusion) we will temporarily denote by $F$. Applying Construction 4.6.4.12 to the inclusion map $\{ F \} \hookrightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$, we obtain a monomorphism of simplicial sets $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} $, which induces a monomorphism

\[ u: \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \{ F\} . \]

Show that the slice diagonal morphism $\delta _{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ of Construction 4.6.4.12 factors (uniquely) through $u$. In particular, $\delta _{/f}$ determines a morphism of simplicial sets $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$. Similarly, the coslice diagonal morphism $\delta _{f/}$ induces a morphism of simplicial sets $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{F/} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$.