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

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
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 S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ R.X ◁
        coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫
      TensorBimod.actLeft (P.tensorBimod Q) L ≫
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                  coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
              ⋯)
          ⋯ =
    R.X ◁
        coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫
      R.X ◁
          coequalizer.desc
            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                  ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
              coequalizer.desc
                ((α_ P.X Q.X L.X).hom ≫
                  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                    coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                      ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
                ⋯)
            ⋯ ≫
        TensorBimod.actLeft P (Q.tensorBimod L)

Tactic:
slice_lhs 1 2 => rw [TensorBimod.whiskerLeft_π_actLeft]

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
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 S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ ((α_ R.X (P.tensorBimod Q).X L.X).inv ≫
        (P.tensorBimod Q).actLeft ▷ L.X ≫
          coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
            ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft)) ≫
      coequalizer.desc
        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.X).hom ≫
              P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                  ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
            ⋯)
        ⋯ =
    R.X ◁
        coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫
      R.X ◁
          coequalizer.desc
            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                  ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
              coequalizer.desc
                ((α_ P.X Q.X L.X).hom ≫
                  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                    coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                      ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
                ⋯)
            ⋯ ≫
        TensorBimod.actLeft P (Q.tensorBimod L)

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
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 S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (tensorLeft R.X).map
        (coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft)) ≫
      TensorBimod.actLeft (P.tensorBimod Q) L ≫
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                  coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
              ⋯)
          ⋯ =
    (tensorLeft R.X).map
        (coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft)) ≫
      R.X ◁
          coequalizer.desc
            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                  ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
              coequalizer.desc
                ((α_ P.X Q.X L.X).hom ≫
                  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                    coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                      ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
                ⋯)
            ⋯ ≫
        TensorBimod.actLeft P (Q.tensorBimod L)

Tactic:
rw [tensorLeft_map]

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
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 S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ R.X ◁
        coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫
      TensorBimod.actLeft (P.tensorBimod Q) L ≫
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                  coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
              ⋯)
          ⋯ =
    R.X ◁
        coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
          ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫
      R.X ◁
          coequalizer.desc
            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                  ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
              coequalizer.desc
                ((α_ P.X Q.X L.X).hom ≫
                  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                    coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                      ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
                ⋯)
            ⋯ ≫
        TensorBimod.actLeft P (Q.tensorBimod L)