nLab
faithful functor

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

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

is one-to-one.

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