Let V be a module of A , and P i j = ϕ ϕ i j ( P ) . Let P : V ⊗ ⊗ V → → V ⊗ ⊗ V be the linear map satisfying P ( x ⊗ ⊗ y ) = y ⊗ ⊗ x for all x , y ∈ ∈ V . The Yang–Baxter equation then has the following alternate form in terms of R ˇ ˇ ( u , u ′ ) = P ∘ ∘ R ( u , u ′ ) on V ⊗ ⊗ V .