Definition 4.7.2.  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 ∈ C. We define
F(X)
by:
(C → DA ↦ YA(f : B → C) ↦ gB,C,f)(X) := YX  (YA ∈ D for each A ∈ C, gB,C,f : YB → YC is a morphism for each B, C ∈ C and f : B → C with suitable conditions)