Kerodon

Example 1.2.2.10. The construction $[n] \mapsto \Delta ^{n}$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to the category $\operatorname{Set_{\Delta }}= \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set})$ of simplicial sets (this is the Yoneda embedding for the simplex category $\operatorname{{\bf \Delta }}$). We regard this functor as a cosimplicial object of $\operatorname{Set_{\Delta }}$, which we denote by $\Delta ^{\bullet }$. Applying Variant 1.2.2.8 to this cosimplicial object, we obtain a functor from the category of simplicial sets to itself, which is canonically isomorphic to the identity functor $\operatorname{id}_{ \operatorname{Set_{\Delta }}}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ (see Proposition 1.1.0.12).