# PlanetPhysics/Fully Faithful Functor 2

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Let ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$ be two categories and


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