#
nLab

faithful functor

### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Definition

A functor $F: C \to D$ from the category $C$ to the category $D$ is *faithful* if for each pair of objects $x, y \in C$, the function

$F : C(x,y) \to D(F(x), F(y))$

between hom sets is injective.

More abstractly, we may say a functor is faithful if it is $2$-surjective – or loosely speaking, ‘surjective on equations between given morphisms’.

See also faithful morphism for a generalization to an arbitrary 2-category.