Total functional relations define morphisms ($R\hookrightarrow A\times B⟹f:A\to B$)

In a regular category, a relation $R\hookrightarrow A\times B$ is total if its projection to $A$ is a regular epimorphism, and single-valued if two pairs with the same first component have the same second component. The latter makes that projection a monomorphism. A regular epimorphism that is monic is invertible, so $f={\pi }_B{\pi }_A^{-1}$ has graph exactly $R$. No axiom of choice or set-based element selection is used.