Definition 2

Let $(X_\iota)_{\iota \in I}$, $(Y_\iota)_{\iota \in I}$ be two families of sets, and let $(g_\iota)_{\iota \in I}$ be a family of functions with the same index set I such that $g_\iota$ is a mapping of $X_\iota$ into $Y_\iota$ for each $\iota \in I$. For each $f \in \prod_{\iota \in I} X_\iota$ let $u_f$ be the graph of the function $\iota = g_\iota(f(\iota))$ $(\iota \in I)$, which is an element of $\prod_{\iota \in I} Y_\iota$. The mapping $f \to u_f$ of $\prod_{\iota \in I} X_\iota$ into $\prod_{\iota \in I} Y_\iota$ is called the canonical extension (or simply the extension) to products of the family of mappings $(g_\iota)_{\iota \in I}$; it is also sometimes called the product of the family of mappings $(g_\iota)_{\iota \in I}$.

When the index notation is used, the product of the family $(g_\iota)_{\iota \in I}$ is the function $(x_\iota)_{\iota \in I} \to (g_\iota(x_\iota))_{\iota \in I}$; it is sometimes denoted by $(g_\iota)_{\iota \in I}$.

If $I = \{\alpha, \beta\}$, where $\alpha$ and $\beta$ are distinct, the extension to products of the family of mappings $(g_\iota)_{\iota \in I}$ is just $\psi \circ (g_\alpha \times g_\beta) \circ \varphi$, where $\varphi$ denotes the canonical mapping of $\prod_{\iota \in I} X_\iota$ onto $X_\alpha \times X_\beta$ (no. 3) and $\psi$ the canonical mapping of $Y_\alpha \times Y_\beta$ onto $\prod_{\iota \in I} Y_\iota$.