Definition 2.1.1.4.  Let S, X, Y be sets, φ:X ↔ Y be a bijection, :S × X → X, :S × Y → Y be operations. We define
φ
by:
(S × X → X(s, x) ↦ as,x)φ(S × Y → Y(s, y) ↦ bs,y)  :⇔  ∀ s ∈ S, x ∈ X : φ(as,x) = bs,φ(x)  (as,x ∈ X for each s ∈ S and x ∈ X, bs,y ∈ Y for each s ∈ S and y ∈ Y)  ⇔  ∀ s ∈ S, x ∈ X : φ(sx) = sφ(x)  ⇔  φ ∘  =  ∘ (idS × φ)
Equivalence.  No proof.