[[!redirects full functor]] [[!redirects faithful functor]] ## Idea ## ## 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** ## See also ## [[Category theory]] [[functor]] [[equivalence of precategories]] ## References ## [[HoTT Book]] category: category theory