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Remark Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the slice and coslice diagonal morphisms

\[ \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]

are monomorphisms of simplicial sets. This follows from Remark, together with the observation that for every simplicial set $J$, the comparison maps

\[ c_{J,K}: J \diamond K \twoheadrightarrow J \star K \quad \quad c_{K,J}: K \diamond J \twoheadrightarrow K \star J \]

are epimorphisms (see Exercise