-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
theorem hom_left_act_hom' :
    (P.tensorBimod (regular S)).actLeft ≫ hom P = (R.X ◁ hom P) ≫ P.actLeft

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)
⊢ (tensorLeft R.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 (tensorLeft R.X) (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)).inv ≫
          colimMap
            (parallelPairHom (R.X ◁ P.actRight ▷ S.X) (R.X ◁ ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)) (P.actRight ▷ S.X)
              ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)
              ((α_ R.X (P.X ⊗ S.X) S.X).inv ≫ (α_ R.X P.X S.X).inv ▷ S.X ≫ P.actLeft ▷ S.X ▷ S.X)
              ((α_ R.X P.X S.X).inv ≫ P.actLeft ▷ S.X) ⋯ ⋯)) ≫
        coequalizer.desc P.actRight ⋯ =
    (tensorLeft R.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)) ≫
      R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft

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

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.actLeft P (regular S) ≫ hom P = R.X ◁ hom P ≫ P.actLeft

Tactic:
dsimp [hom, TensorBimod.actLeft, 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 (tensorLeft R.X) (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)).inv ≫
        colimMap
          (parallelPairHom (R.X ◁ P.actRight ▷ S.X) (R.X ◁ ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)) (P.actRight ▷ S.X)
            ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul)
            ((α_ R.X (P.X ⊗ S.X) S.X).inv ≫ (α_ R.X P.X S.X).inv ▷ S.X ≫ P.actLeft ▷ S.X ▷ S.X)
            ((α_ R.X P.X S.X).inv ≫ P.actLeft ▷ S.X) ⋯ ⋯)) ≫
      coequalizer.desc P.actRight ⋯ =
    R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft

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).inv ≫ P.actLeft ▷ S.X) ≫ P.actRight =
    R.X ◁ coequalizer.π (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ≫
      R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft

Tactic:
slice_lhs 2 3 => rw [middle_assoc]

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).inv ≫ (α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight ≫ P.actLeft =
    R.X ◁ coequalizer.π (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ≫
      R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft