> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-functors/faithful.md).

# 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