A name and notation for a category

Question

Is there a name and a standard notation for the category $C$ defined as follows:

• objects of $C$ are all small "objects" (small sets in ZF, because in ZF all objects are sets);

• morphisms of $C$ from $a$ to $b$ are all small functions such that $f(a)=b$.

Answer 1

I would probably call this something like "the subcategory of epimorphisms in Set". If I were to use it a lot, especially with other categories, I would probably name it something like Epi(Set).