Construction 3.5.5.1 (Geometric Realization as a Simplicial Functor). Let $X$ and $Y$ be simplicial sets and let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Fun}(X,Y)$, which we identify with a morphism $\Delta ^ n \times X \rightarrow Y$. By virtue of Corollary 3.5.2.2, the geometric realization of $\sigma $ can be identified with a continuous function

which we can view as an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{Top}}(|X|,|Y|)_{\bullet }$ parametrizing continuous functions from $X$ to $Y$ (see Example 2.4.1.5). This construction is compatible with face and degeneracy operators, and therefore determines a morphism of simplicial sets $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( |X|, |Y| )_{\bullet }$. Allowing $X$ and $Y$ to vary, we obtain a simplicial structure on the geometric realization functor $| \bullet |: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}$.