Faithful functor

https://en.wikipedia.org/wiki/Full_and_faithful_functors

A full functor is a functor that is surjective when restricted to each set of morphisms that have a given source and target.

A faithful functor is a functor that is injective when restricted to each set of morphisms that have a given source and target.

A fuly faithful functor is a functor that is bijective when restricted to each set of morphisms that have a given source and target.

Formal definitions

Explicitly, if C and D are (locally small) categories, and F : C → D is a functor from C to D. Then, for every pair of objects X and Y in C, the functor F induces a function

Fx,ʏ : HOMᴄ (X,Y) -> HOMᴅ (F(X), F(Y))

$$F_{X,Y} \colon \mathrm{Hom}_ {\mathcal C}(X,Y) \rightarrow \mathrm{Hom}_ {\mathcal D}(F(X),F(Y))$$

∀XY ∈ C, the functor F is

• faithful if Fx,ʏ is injective

• full if Fx,ʏ is surjective

• fully faithful if Fx,ʏ is bijective