PlanetPhysics/Fully Faithful Functor 2

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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.