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

1. 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.

2. If these maps are all bijective then $F$ is called fully faithful.

3. 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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