If $ \mathscr{A} $ is small then the class of objects of $ \mathscr{A} $ is small too, since objects correspond one - to - one with identity maps. A category $ \mathscr{A} $ is locally small if for each $ A, B \in \mathscr{A} $ , the class $ \mathscr{A}(A, B) $ is small. (So, small implies locally small.)