Remark 2

Let $\alpha$, $\beta$, $\gamma$ be three objects, no two of which are equal (three such objects exist; for example, $\emptyset$, $\{\emptyset\}$, $\{\{\emptyset\}\}$), and let A, B, C be sets. Consider the functional graph $\{(\alpha, A), (\beta, B), (\gamma, C)\}$, i.e., the family of sets $(X_\iota)_{\iota \in \{\alpha, \beta, \gamma\}}$ such that $X_\alpha = A$, $X_\beta = B$, $X_\gamma = C$. To the partition of $\{\alpha, \beta, \gamma\}$ formed by the two sets $\{\alpha, \beta\}$ and $\{\gamma\}$ there corresponds a canonical bijection of $\prod_{\iota \in \{\alpha, \beta, \gamma\}} X_\iota$ onto the product

and hence a bijection (again called canonical) of $\prod_{\iota \in \{\alpha, \beta, \gamma\}} X_\iota$ onto $A \times B \times C$ (§ 2, no. 2) which transforms each graph $f \in \prod_{\iota \in \{\alpha, \beta, \gamma\}} X_\iota$ into the element $(f(\alpha), f(\beta), f(\gamma))$ of $A \times B \times C$. By Proposition 4 we may therefore put any two of the six sets $A \times B \times C$, $B \times C \times A$, $C \times A \times B$, $B \times A \times C$, $A \times C \times B$, $C \times B \times A$ in one-to-one correspondence.