fully faithful functor

Definition 0.1   Let $ \mathcal{A}$ and $ \mathcal{B}$ be two categories and let $ F: \mathcal{A} \to \mathcal{B}$ be a functor. $ F$ is said to be a fully faithful functor if it is an isomorphism on every set $ Hom(-,-)$ of morphisms, and that it is essentially surjective if for every object $ X \in \mathcal{B}$ , there is some $ Y \in \mathcal{A}$ such that $ X$ and $ F(Y)$ are isomorphic.



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