Homotopy Type Theory fully faithful (Rev #3)

Definition

A functor $F : A \to B$ is faithful if for all $a,b : A$, the function

$F_{a,b} : hom_A(a,b) \to hom_B(F a, F b)$

is injective?, and full if for all $a,b : A$ this function is surjective?. If it is both then $F$ is fully faithful