---
tag: 3GZ99NCX22
kind: definition
name: Full and faithful functors
msc: [18A22]
defines: GRG8WEBSH5
uses: [28NQ0KRHD3, V3BFGKBJF7, G5S1M9, 0JYTG5K9Y4, 09NWETVTNY, 01RVCMSN5N]
sources:
  - url: https://en.wikipedia.org/wiki/Full_and_faithful_functors
---

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]].
For objects $A$, $B$ of $\mathcal{C}$, let $F_{A, B} \colon \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{D}}(F(A), F(B))$ be the map $f \mapsto F(f)$.

1. $F$ is *faithful* if $F_{A, B}$ is [[injective|09NWETVTNY]] for all objects $A$, $B$.
2. $F$ is *full* if $F_{A, B}$ is [[surjective|01RVCMSN5N]] for all objects $A$, $B$.
3. $F$ is *fully faithful* if $F_{A, B}$ is bijective for all objects $A$, $B$.

## Remarks

A faithful functor may still identify two morphisms with different domains or codomains, and it may send different objects to the same object.
Example: the forgetful functor from [[groups|G5S1M9]] to sets is faithful, since two [[homomorphisms|0JYTG5K9Y4]] with the same underlying map are equal.
It is not full: the constant map $\Z \to \Z$ with value $1$ is a function between the underlying sets but not a homomorphism.
The inclusion of a [[full subcategory|V3BFGKBJF7]] is fully faithful.
Non-example: the functor from groups to the category with one object and one morphism is not faithful.
The two homomorphisms $\Z \to \Z$ given by $x \mapsto 0$ and $x \mapsto x$ are sent to the same morphism.
A fully faithful functor reflects isomorphisms: if $F(f)$ is an isomorphism, then so is $f$.