A cone is a natural transformation, a member of Nat(ΔV, F), the set of natural transformation between two functors, ΔV and F. A morphism h is a member of the home set Hom(V, U), the set of morphisms between two objects. So we can say that if U is a limit of F then there is an isomorphism between those two sets:
Nat(ΔV, F) ~ Hom(V, U)