The Stacks project

Remark 14.26.5. Let $\mathcal{C}$ be any category. Suppose two morphisms $a, a' : U \to V$ of simplicial objects are homotopic. Then for any morphism $b : V \to W$ the two maps $b \circ a, b \circ a' : U \to W$ are homotopic. Similarly, for any morphism $c : X \to U$ the two maps $a \circ c, a' \circ c : X \to V$ are homotopic. In fact the maps $b \circ a \circ c, b \circ a' \circ c : X \to W$ are homotopic. Namely, if the maps $h_{n, i} : U_ n \to V_ n$ define a homotopy from $a$ to $a'$ then the maps $b \circ h_{n, i} \circ c$ define a homotopy from $b \circ a \circ c$ to $b \circ a' \circ c$. In this way we see that we obtain a new category $\text{hSimp}(\mathcal{C})$ with the same objects as $\text{Simp}(\mathcal{C})$ but whose morphisms are homotopy classes of of morphisms of $\text{Simp}(\mathcal{C})$. Thus there is a canonical functor

which is essentially surjective and surjective on sets of morphisms.