Warning 4.3.5.11. Recall that, if $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories, then the slice category $\operatorname{\mathcal{C}}_{/F}$ can be defined as the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}}) } \{ F\} $ (see Remark 4.3.1.11). In the setting of simplicial sets, our definition is somewhat different. Nevertheless, to any morphism of simplicial sets $F: K \rightarrow \operatorname{\mathcal{C}}$, one can associate a comparison map
which we will refer to as the slice diagonal morphism (see Construction 4.6.4.13). This map has the following features:
When $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, the morphism $\delta _{/F}$ is an isomorphism of simplicial sets.
When $\operatorname{\mathcal{C}}$ is an $\infty $-category, the morphism $\delta _{/F}$ is an equivalence of $\infty $-categories (Theorem 4.6.4.17).
When $\operatorname{\mathcal{C}}= \operatorname{Sing}_{\bullet }(X)$ is the singular simplicial set of a topological space $X$, the morphism $\delta _{/F}$ does not coincide with the isomorphism constructed in Example 4.3.5.9 (however, they are naturally homotopic).
The morphism $\delta _{/F}$ is usually not an isomorphism of simplicial sets (see Warning 4.3.3.33).