Definition 2.1.1.5.  Let S, X, Y be sets, φ:X ↔ Y be a bijection, :X × S → X, :Y × S → Y be operations. We define
φ
by:
(X × S → X(x, s) ↦ ax,s)φ(Y × S → Y(y, s) ↦ by,s)  :⇔  ∀ x ∈ X, s ∈ S : φ(ax,s) = bφ(x),s  (ax,s ∈ X for each x ∈ X and s ∈ S, by,s ∈ Y for each y ∈ Y and s ∈ S)  ⇔  ∀ x ∈ X, s ∈ S : φ(xs) = φ(x) s  ⇔  φ ∘  =  ∘ (φ × idS)
Equivalence.  No proof.