$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $X$, $Y$, and $Z$ be topological spaces. Then the datum of a continuous function $X \star Y \rightarrow Z$ is equivalent to the datum of a triple $(f_ X, f_ Y, h)$, where $f_ X: X \rightarrow Z$ and $f_ Y: Y \rightarrow Z$ are continuous functions and $h: X \times [0,1] \times Y \rightarrow Z$ is a homotopy from $f_{X} \circ \pi _{X}$ to $f_{Y} \circ \pi _{Y}$; here $\pi _{X}: X \times Y \rightarrow X$ and $\pi _{Y}: X \times Y \rightarrow Y$ denote the projection maps.