In this [posting][1] what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "[MK][2]-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this [posting][3]), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:

**Axiom of Small Class Choice :**$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$ 

In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain. 

A class is said to be *small* if and only if it is strictly smaller in cardinality than the universe $V$ of all sets. 

$^*$ *Set Replacement* is the assertion: "replacement classes from sets, are sets".
 
>Questions

> 1. Are there models of the "base theory + small class choice" in which there exist small proper classes?

> 2. If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory? I mean are there models satisfying the base theory in which choice over sets + collection is satisfied but small class choice fails?


  [1]: https://mathoverflow.net/questions/351707/is-proper-class-choice-equivalent-to-global-choice/351711#351711
  [2]: https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory
  [3]: https://math.stackexchange.com/questions/3533971/is-set-sized-choice-over-proper-classes-weaker-than-global-choice/3534052?noredirect=1#comment7270161_3534052