Remark 1

Let $\alpha$, $\beta$ be two distinct objects and let $(J_\lambda)_{\lambda \in \{\alpha, \beta\}}$ be a partition of I into two sets $J_\alpha$, $J_\beta$. We thus obtain a one-to-one mapping (again called canonical) of the product $\prod_{\iota \in I} X_\iota$ onto $\left( \prod_{\iota \in J_\alpha} X_\iota \right) \times \left( \prod_{\iota \in J_\beta} X_\iota \right)$ by forming the composition of the canonical mapping of $\prod_{\lambda \in \{\alpha, \beta\}} \left( \prod_{\iota \in J_\lambda} X_\iota \right)$ onto $\left( \prod_{\iota \in J_\alpha} X_\iota \right) \times \left( \prod_{\iota \in J_\beta} X_\iota \right)$ and the canonical mapping of $\prod_{\iota \in I} X_\iota$ onto $\prod_{\lambda \in \{\alpha, \beta\}} \left( \prod_{\iota \in J_\lambda} X_\iota \right)$. If $X_\iota$ is a set consisting of a single element for each $\iota \in J_\beta$, then $\mathrm{pr}_{J_\alpha}$ is a bijective mapping of $\prod_{\iota \in I} X_\iota$ onto $\prod_{\iota \in J_\alpha} X_\iota$.