Implication Details

Claim: If a category is concretizable, then it is locally essentially small.

Proof: If $U : \C \to \Set$ is a faithful functor, for every pair of objects $A,B$, the collection $\Hom(A,B)$ embeds into the set $\Hom(U(A),U(B))$, so that it is isomorphic to a set.