-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
h :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N' ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
    ((M.X ◁ f.hom ≫
            colimit.ι (parallelPair (M.actRight ▷ N'.X) (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft))
              WalkingParallelPair.one) ▷
          P.X ≫
        colimit.ι
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft))
          WalkingParallelPair.one :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
        colimit
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
    (Iso.hom (α_ M.X N.X P.X) ≫
        CategoryStruct.comp (Y :=
          TensorBimod.X M
            { X := TensorBimod.X N' P, actLeft := TensorBimod.actLeft N' P, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight N' P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
              Bimod X Z })
          (M.X ◁
                (f.hom ▷ P.X ≫
                  colimit.ι (parallelPair (N'.actRight ▷ P.X) (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft))
                    WalkingParallelPair.one) ≫
              colimit.ι
                (parallelPair (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N' P))
                  (M.actRight ▷ TensorBimod.X N' P)
                  (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P))
                WalkingParallelPair.one :
            tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
              TensorBimod.X M
                { X := TensorBimod.X N' P, actLeft := TensorBimod.actLeft N' P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N' P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod X Z })
          (AssociatorBimod.inv M N' P : (M.tensorBimod (N'.tensorBimod P)).X ⟶ ((M.tensorBimod N').tensorBimod P).X) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
        colimit
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
⊢ Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N' ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
    ((M.X ◁ f.hom ≫
            colimit.ι (parallelPair (M.actRight ▷ N'.X) (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft))
              WalkingParallelPair.one) ▷
          P.X ≫
        colimit.ι
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft))
          WalkingParallelPair.one :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
        colimit
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
    (Iso.hom (α_ M.X N.X P.X) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
          (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft))
          (CategoryStruct.comp (Y :=
            TensorBimod.X M
              { X := TensorBimod.X N' P, actLeft := TensorBimod.actLeft N' P, one_actLeft := ⋯, left_assoc := ⋯,
                actRight := TensorBimod.actRight N' P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                Bimod X Z })
            (M.X ◁
                  colimMap
                    (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)
                      (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯) ≫
                colimit.ι
                  (parallelPair (X :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N' P))
                    (M.actRight ▷ TensorBimod.X N' P)
                    (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P))
                  WalkingParallelPair.one :
              (parallelPair (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N P))
                      (M.actRight ▷ TensorBimod.X N P)
                      (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P)).obj
                  WalkingParallelPair.one ⟶
                colimit
                  (parallelPair (X :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N' P))
                    (M.actRight ▷ TensorBimod.X N' P)
                    (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P)))
            (AssociatorBimod.inv M N' P :
              (M.tensorBimod (N'.tensorBimod P)).X ⟶ ((M.tensorBimod N').tensorBimod P).X)) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) ⟶
        colimit
          (parallelPair (TensorBimod.actRight M N' ▷ P.X)
            (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))

Tactic:
slice_rhs 2 3 => rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.Limits.parallelPairHom_app_one