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(x ∗ s) = φ(x) ⋆ s  ⇔  φ ∘ ∗ = ⋆ ∘ (φ × idS)
Equivalence.  No proof.