Definition 4.2.9. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor.

We say $F$ is

*faithful*if for any objects $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the map\[ F : \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(F(x), F(y)) \]is injective.

If these maps are all bijective then $F$ is called

*fully faithful*.The functor $F$ is called

*essentially surjective*if for any object $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ there exists an object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\mathcal{B}$.

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