Implication Details

Claim: If a category is gaunt and is locally small and is trivial, then it is small.

Proof: The category is trivial and therefore inhabited. Choose an object $A$. For every object $B$, since the category is trivial, there is an isomorphism $f : A \to B$. Since the category is gaunt, $f$ is the identity. In particular, $A = B$. Therefore, there is exactly one object $A$. The collection $\Hom(A,A)$ is a set since the category is locally small. Thus, the category is small.