One version of the Axiom of Collection says that any surjection $A\to B$ from a class $A$ to a set $B$ is factored through by some surjection $C\to B$ where $C$ is a set.

Note that assuming $B$ is a set, the axiom of replacement ensures that $C$ is a set if and only if each fiber of $C\to B$ is a set (i.e. the map $C\to B$ is "small" in the sense of algebraic set theory). Thus it is natural to consider the following "class-collection axiom": any surjection $A\to B$ of classes is factored through by some surjection $C\to B$ whose fibers are all sets.

Has this "class-collection axiom" been studied at all? Assuming classical logic, it seems to follow from the axiom of foundation in the same way that collection follows from replacement and foundation: let the fiber of $C$ over $b\in B$ consist of all those $a\in A$ lying over $b$ and of minimal rank. Can it be proven in any intuitionistic context?