PlanetPhysics/Fully Faithful Functor 2

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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.