Definition 4.2.  Let u ∈ ℕ, C be a/an u-small category. We define
Ob(C)
by:
Ob([O(MA,B)A,B ∈ O(iC)C ∈ O(∘D,E,F)D,E,F ∈ O]u) := O  (O is a set, MA,B is a set for each A, B ∈ O, iC ∈ MC,C for each C ∈ O, ∘D,E,F : ME,F × MD,E → MD,F is an operation for each D, E, F ∈ O with suitable conditions)
We write “let X ∈ C” for “let X ∈ Ob(C).”
Remarks.

This definition retrieves a representative set of objects from a category (which is actually an equivalence class of categories), and enables typical abuse of notation.