Definition:Faithful Functor

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

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

Then $F$ is faithful 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 an injection.

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

Also see

 * Definition:Injective on Morphisms
 * Definition:Full Functor
 * Definition:Embedding of Categories