Starting with Morse-Kelley set theory $\sf MK$:
Can we consistently make the following additions to $\sf MK$:
Add a primitive total unary function $F^\phi$ for every predicate symbol $\phi$ of the language (definable or primitive).
For convinience elements would be represented in lower case, while classes are represented in upper case. [the language is mono-sorted, so this is pseuo-sorting]. Add the following schema:
Universal equivalence relations: if $\phi$ is a predicate symbol, then:
$\phi \text{ is equivalence relation } \ \\ \forall A \forall x \exists B : \phi(A,B) \land x \in B \\ \implies \\\forall A \exists k: k=F^\phi(A) \ \\ \forall A \forall B: F^\phi(A) = F^\phi(B) \iff \phi(A,B) $