If we add to the language of ZF a one place predicate symbol $\frak Can$, standing for is Canonical, then add to axioms of $\sf ZF$ the following axiom:
Canonicity: $\forall R: R \text { is a relation } \to \exists! \frak X: Can(X)$$ \land \frak X$$ \, \approx R$
Where $\approx$ signify isomorphic to; defined as:
$S \approx R \iff \exists f: dom(S) \cup rng(S) \to dom(R) \cup rng(R) \land f \text{ is bijective } \land \\\forall a,b: \langle a,b \rangle \in S \Leftrightarrow \langle f(a),f(b) \rangle \in R$
; while is a relation means being a subset of a cartesian product of two sets.
Now the above is clearly a theorem of $\sf ZF +GC$, but is the opposite true?
That is: is global choice $\sf GC$, or even choice $\sf C$, provable in $\sf ZF + Canonicity$?