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Remark Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. For every simplicial set $J$, composition with the slice diagonal $\delta _{/F}$ of Construction determines a map of sets

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( J, \operatorname{\mathcal{C}}_{/F} ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( J, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} ). \]

Under the bijection of Remark, this identifies with the map

\[ \operatorname{Hom}_{(\operatorname{Set_{\Delta }})_{K/} }( J \star K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{(\operatorname{Set_{\Delta }})_{K/} }( J \diamond K, \operatorname{\mathcal{C}}) \]

given by precomposition with the comparison map $c_{J,K}: J \diamond K \twoheadrightarrow J \star K$ of Notation