If we add to the language of $\sf ZF$, a unary predicate $C$, and axiomatize that for every equinumerouse sets $X,Y$ there exists a unique bijection $f$ from $X$ to $Y$ such that $C(f)$.
Is this prinicple equivalent to choice?
If not, then is it equivalent to some weak form of choice?
Note: the symbol $C$ is to be usable in instances of Separation and Replacement.
As explained in the comments by Asaf and myself: We show that we can canonically choose an element in any non-empty set, implying the axiom of global choice. Let $A$ be a non-empty set. The set $A \times \omega$ is in bijection with $(A \times \omega) \cup \{A \times \omega\}$, because it is a Dedekind-infinite set. Choosing the bijection $f : (A \times \omega) \cup \{A \times \omega\} \to A \times \omega$ such that $C(f)$, we have some element $(a,n) = f(A \times \omega)$. The canonical element in $A$ is then $a$.