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Construction 4.6.4.12 (The Slice Diagonal Morphism). Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $c: \operatorname{\mathcal{C}}_{/F} \diamond K \rightarrow \operatorname{\mathcal{C}}_{/F} \star K$ be the comparison morphism of Notation 4.5.8.3. By virtue of Remark 4.6.4.8, the composite map

\[ \operatorname{\mathcal{C}}_{/F} \diamond K \xrightarrow {c} \operatorname{\mathcal{C}}_{/F} \star K \rightarrow \operatorname{\mathcal{C}} \]

determines a morphism of simplicial sets $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we will refer to as the slice diagonal morphism. Similarly, the composition

\[ K \diamond \operatorname{\mathcal{C}}_{F/} \rightarrow K \star \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}} \]

determines a morphism of simplicial sets $\delta _{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$, which we will refer as the coslice diagonal morphism.