category theory

# 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

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

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’.