Example 3.1.5.5. Let $X_{}$ be a simplicial set and let $Y$ be a topological space. Suppose we are given a pair of continuous functions $f_0, f_1: | X_{} | \rightarrow Y$, corresponding to morphisms of simplicial sets $f'_0, f'_1: X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$. Let $h: [0,1] \times | X_{} | \rightarrow Y$ be a continuous function satisfying $f_0 = h|_{ \{ 0\} \times |X_{}|}$ and $f_1 = h|_{ \{ 1\} \times | X_{} |}$ (that is, a homotopy from $f_0$ to $f_1$ in the category of topological spaces). Then the composite map

classifies a morphism of simplicial sets $h': \Delta ^1 \times X_{} \rightarrow \operatorname{Sing}_{\bullet }(Y)$, which is a homotopy from $f'_0$ to $f'_1$ (in the sense of Definition 3.1.5.2). We will show later that $\theta $ is a homeomorphism of topological spaces (Corollary 3.6.2.2), so every homotopy from $f_0$ to $f_1$ arises in this way. In other words, the construction $h \mapsto h'$ induces a bijection