Definition:Full Functor

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Let $\mathbf C$ and $\mathbf D$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a covariant functor.

Then $F$ is full iff for all objects $C_1, C_2$ of $\mathbf C$:

$F: \operatorname{Hom}_{\mathbf C} \left({C_1, C_2}\right) \to \operatorname{Hom}_{\mathbf D} \left({F C_1, F C_2}\right), \ f \mapsto F f$

is a surjection.

Here $\operatorname{Hom}$ signifies a hom class.

Also see