Faithful functor - Knowlpedia

Let $F:\mathcal C\to\mathcal D$ be a functor. It is faithful if, for every pair of objects $X,Y$ in $\mathcal C$ , the induced map

\operatorname{Hom}_{\mathcal C}(X,Y)\longrightarrow \operatorname{Hom}_{\mathcal D}(F(X),F(Y)),\qquad f\longmapsto F(f)

is injective.

Interpretation

A faithful functor can identify distinct objects, but it never identifies two morphisms having the same source and target.

Example

The forgetful functor from groups to sets is faithful: a group homomorphism is determined by its underlying function.