Kerodon

Exercise 1.4.3.4. Let $\pi : [0,1] \times [0,1] \rightarrow | \Delta ^2 |$ denote the continuous function given by the formula $\pi (s,t) = ( 1-s, (1-t)s, ts )$. For any topological space $X$, the construction $\sigma \mapsto \sigma \circ \pi $ determines a map from the set $\operatorname{Sing}_{2}(X)$ of singular $2$-simplices of $X$ to the set of all continuous functions $H: [0,1] \times [0,1] \rightarrow X$. Show that, if $f,g: [0,1] \rightarrow X$ are continuous paths satisfying $f(0) = g(0)$ and $f(1) = g(1)$, then the construction $\sigma \mapsto \sigma \circ \pi $ induces a bijection from the set of homotopies from $f$ to $g$ (in the sense of Definition 1.4.3.1) to the set of continuous functions $H$ satisfying the requirements of Example 1.4.3.3.