The co- and contravariant hom-functors $\mathrm{Hom}(B,-)$ and $\mathrm{Hom}(-,B)$ are maybe the most natural functors. While forgetful functors are other examples of covariant set-valued functors, covariant functors very often have to do with function spaces. (Once we pass from presheaves to sheaves by adding some more “topological requirements”, this becomes a theorem: sheaves can always be viewed as evaluating to collections of function spaces.)

Wikipedia: Yoneda lemma, Functor category

Functor category

Cat

Set, Opposite category