Proposition 11

Let $(X_\iota)_{\iota \in I}$, $(Y_\iota)_{\iota \in I}$, $(Z_\iota)_{\iota \in I}$ be three families of sets and let $(g_\iota)_{\iota \in I}$, $(g'_\iota)_{\iota \in I}$ be two families of functions, all having the same index set, such that $g_\iota$ is a mapping of $X_\iota$ into $Y_\iota$ and $g'_\iota$ a mapping of $Y_\iota$ into $Z_\iota$, for each $\iota \in I$. Let $g$ and $g'$ be the extensions of the families $(g_\iota)_{\iota \in I}$ and $(g'_\iota)_{\iota \in I}$ to products. Then the extension of the family $(g'_\iota \circ g_\iota)_{\iota \in I}$ to products is equal to $g' \circ g$.

This follows immediately from Definition 2.