Kerodon

Example 3.2.1.9. Let $(X,x)$ and $(Y,y)$ be pointed topological spaces, and let $h: [0,1] \times X \rightarrow Y$ be a continuous function satisfying $h(t,x) = y$ for $0 \leq t \leq 1$, which we regard as a pointed homotopy from $f_0 = h|_{ \{ 0\} \times X}$ to $f_1 = h|_{ \{ 1\} \times X}$. Then $h$ determines a homotopy between the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(f_0), \operatorname{Sing}_{\bullet }(f_1): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(Y)$: this follows by applying Example 3.2.1.8 to the composite map $[0,1] \times | \operatorname{Sing}_{\bullet }(X) | \rightarrow [0,1] \times X \xrightarrow {h} Y$.