Definition 4.7.3.  Let u, v ∈ ℕ, C be a/an u-small category, D be a/an v-small category, F : C → D be a functor, X, Y ∈ C, f : X → Y be a morphism. We define
F(f)
by:
(C → DA ↦ DA(g : B → C) ↦ hB,C,g)(f) := hX,Y,f  (DA ∈ D for each A ∈ C, hB,C,g : DB → DC is a morphism for each B, C ∈ C and g : B → C with suitable conditions)