Definition 2.1.1.2.  Let X, Y be sets, φ:X ↔ Y be a bijection, f:X → X, g:Y → Y be functions. We define
fφg
by:
(X → Xx ↦ ax)φ(Y → Yy ↦ by)  :⇔  ∀ x ∈ X : φ(ax) = bφ(x)  (ax ∈ X for each x ∈ X, by ∈ Y for each y ∈ Y)  ⇔  ∀ x ∈ X : φ(f(x)) = g(φ(x))  ⇔  φ ∘ f = g ∘ φ
Equivalence.  No proof.