Proposition 3

Let X be a topological space and let $(Y_j)_{j\in J}$ be a family of topological spaces. The mapping of $[X;\prod_{j\in J}Y_j]$ into the product set $\prod_{j\in J}[X; Y_j]$ defined by $\varphi \mapsto$ ([pr$_j]\circ \varphi$ )$_{j\in J}$ is bijective.

Surjectivity results immediately from I, p. 25, prop. 1. Let us prove injectivity. Let $f$ and $g$ be continuous mappings of X into $\prod_{j\in J}Y_j$. For each $j\in J$, put $f_j=$ pr$_j\circ f$ and $g_j=$ pr$_j\circ g$; suppose $f_j$ homotopic to $g_j$ and let $\sigma_j$ be a homotopy joining $f_j$ to $g_j$. The mapping $\sigma = (\sigma_j)$ of $X\times \mathbf{I}$ into $\prod_{j\in J}Y_j$ is continuous (loc. cit.) ; it is a homotopy joining $f$ to $g$, whence the proposition.