-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
theorem hom_right_act_hom' :
    (P.tensorBimod (regular S)).actRight ≫ hom P = (hom P ▷ S.X) ≫ P.actRight

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (α_ P.X S.X S.X).hom ≫ (α_ P.X S.X S.X).inv ≫ P.actRight ▷ S.X ≫ P.actRight = P.actRight ▷ S.X ≫ P.actRight

Tactic:
rw [Iso.hom_inv_id_assoc]

Result:
no goals

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)).inv ≫
        colimMap
          (parallelPairHom (P.actRight ▷ S.X ▷ S.X) (((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ▷ S.X) (P.actRight ▷ S.X)
            ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)
            ((α_ (P.X ⊗ S.X) S.X S.X).hom ≫ (α_ P.X S.X (S.X ⊗ S.X)).hom ≫ P.X ◁ S.X ◁ S.mul ≫ (α_ P.X S.X S.X).inv)
            ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ⋯ ⋯)) ≫
      coequalizer.desc P.actRight ⋯ =
    coequalizer.desc P.actRight ⋯ ▷ S.X ≫ P.actRight

Tactic:
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (tensorRight S.X).map
        (coequalizer.π
          (P.actRight ▷
            { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.X)
          ((α_ P.X S.X
                { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X).hom ≫
            P.X ◁
              { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.actLeft)) ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)).inv ≫
          colimMap
            (parallelPairHom (P.actRight ▷ S.X ▷ S.X) (((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ▷ S.X) (P.actRight ▷ S.X)
              ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)
              ((α_ (P.X ⊗ S.X) S.X S.X).hom ≫ (α_ P.X S.X (S.X ⊗ S.X)).hom ≫ P.X ◁ S.X ◁ S.mul ≫ (α_ P.X S.X S.X).inv)
              ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ⋯ ⋯)) ≫
        coequalizer.desc P.actRight ⋯ =
    (tensorRight S.X).map
        (coequalizer.π
          (P.actRight ▷
            { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.X)
          ((α_ P.X S.X
                { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X).hom ≫
            P.X ◁
              { X := S.X, actLeft := S.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := S.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.actLeft)) ≫
      coequalizer.desc P.actRight ⋯ ▷ S.X ≫ P.actRight

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ TensorBimod.actRight P (regular S) ≫ hom P = hom P ▷ S.X ≫ P.actRight

Tactic:
dsimp [hom, TensorBimod.actRight, regular]

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)).inv ≫
        colimMap
          (parallelPairHom (P.actRight ▷ S.X ▷ S.X) (((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ▷ S.X) (P.actRight ▷ S.X)
            ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)
            ((α_ (P.X ⊗ S.X) S.X S.X).hom ≫ (α_ P.X S.X (S.X ⊗ S.X)).hom ≫ P.X ◁ S.X ◁ S.mul ≫ (α_ P.X S.X S.X).inv)
            ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ⋯ ⋯)) ≫
      coequalizer.desc P.actRight ⋯ =
    coequalizer.desc P.actRight ⋯ ▷ S.X ≫ P.actRight