**Exercise 5.2.24**

Denote by the full subcategory of whose objects are the epic morphisms. Call a “quotient object” of an isomorphism class of objects in .

**Theorem**:
Let and be epics in .
and are isomorphic in if and only if they induce the same equivalence relation on .

*Proof*:
Suppose and are isomorphic in .
This means there is an isomorphism such that .
Let be the equivalence relation induced by , similarly for .
Suppose that .
In particular .
Then, .
Since is in particular injective, we have .
So .
The converse follows similarly by noting that we have an isomorphism and .
So .

Suppose that . Define a map as follows. Given an , choose an such that (which must exist, since epics in are surjective), and let . This is well defined because, if we were to choose a different element with , we have so and . To see is injective, take . Then we must have an with and but (if it were, would be identical on and ) – it follows that so , so takes distinct values. It’s easy to see that is surjective – given an , take an with , and note that the element maps onto . So is a bijection and thus an isomorphism. To check that it is an isomorphism in , we must ensure that . Indeed, given , is equal to for some element with . But of course because . So, is indeed an isomorphism of epics.

So we have the dual result to that of 5.1.40 – the “quotient objects” of are in correspondence with the equivalence relations on .