-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
theorem hom_right_act_hom' :
    ((regular R).tensorBimod P).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)
⊢ (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight

Tactic:
dsimp

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)
⊢ coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ 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 (regular R) P ≫ 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) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
        colimMap
          (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
            ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
            ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
              (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
            ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
      coequalizer.desc P.actLeft ⋯ =
    coequalizer.desc P.actLeft ⋯ ▷ 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_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight

Tactic:
slice_rhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]

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)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight