-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
A B : Mon.{v₁, u₁} C
M_1 : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y := TensorBimod.X M N₁)
        (Iso.inv (α_ X.X M.X N₁.X) ≫
          M.actLeft ▷ N₁.X ≫ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
        (colimMap
          (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
            (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
            (M.X ◁ f.hom) ⋯ ⋯)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
        (X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
        (X.X ◁
            colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ≫
          TensorBimod.actLeft M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
        (X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) X.X
              (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
        (TensorBimod.actLeft M N₁ ≫
            colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) :
          tensorObj.{v₁, u₁} (𝒞 := inst) X.X (TensorBimod.X M N₁) ⟶ TensorBimod.X M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
        (X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
        (X.X ◁
            colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ≫
          TensorBimod.actLeft M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)

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

Premises used: Bimod.TensorBimod.whiskerLeft_π_actLeft