Proposition 3

If for every mapping $f$ of $\mathrm{B} \times \mathrm{C}$ into A we denote by $\tilde{f}$ the mapping $y \to f(\bullet, y)$ of C into $\mathfrak{F}(\mathrm{B}, \mathrm{A})$, then the function $f \to \tilde{f}$ is a bijection (called the canonical bijection) of $\mathfrak{F}(\mathrm{B} \times \mathrm{C}, \mathrm{A})$ onto $\mathfrak{F}(\mathrm{C}, \mathfrak{F}(\mathrm{B}, \mathrm{A}))$.

Similarly we define a canonical bijection of $\mathfrak{F}(\mathrm{B} \times \mathrm{C}, \mathrm{A})$ onto $\mathfrak{F}(\mathrm{B}, \mathfrak{F}(\mathrm{C}, \mathrm{A}))$. By reason of the one-to-one correspondence between mappings and functional graphs, these bijections give rise to canonical bijections of $\mathrm{A}^{\mathrm{B} \times \mathrm{C}}$ onto $(\mathrm{A}^{\mathrm{B}})^{\mathrm{C}}$ (resp. $(\mathrm{A}^{\mathrm{C}})^{\mathrm{B}}$).