We added a section on epimorphisms of rings to the algebra chapter. Everything is completely straightforward except for the following fact: If A —> B is an epimorphism of rings then |B| ≤ |A|. Since an epimorphism of rings is not necessarily surjective this is not a triviality. We learned this from the exposee by Mazet in the Seminaire Samuel.
You can use this to show that if X —>Y is a monomorphism of schemes then size(X) ≤ size(Y), which is a technical condition on the cardinalities of some sets associated to X and Y. See the chapter on sets.
Here is a consequence: Given a scheme Y there is a set worth of isomorphism classes of monomorphisms X —> Y. I don’t think this is formal since I vaguely remember reading somewhere about a category (maybe spaces up to homotopy?) where such a thing is not true. Leave a comment if you know the correct statement.