-- 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 :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X ⟶
      TensorBimod.X
        { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
        P)
    (colimMap
            (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
              (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom)
              (M.X ◁ f.hom) ⋯ ⋯) ▷
          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 :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.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)))
    (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 })
        (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 })
            (CategoryStruct.comp (Y :=
                coequalizer (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst)
                    (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) N.X) P.X)
                  (M.actRight ▷ N.X ▷ P.X) ((Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X))
                (Iso.inv
                  (PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
                    (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)))
                (coequalizer.desc
                  (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) :
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
                          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)))
                      (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
                          (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) :
                        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)) ⟶
                          coequalizer (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)))
                  ⋯) :
              tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X ⟶ TensorBimod.X M (N.tensorBimod P))
            (colimMap
                (parallelPairHom (M.actRight ▷ TensorBimod.X N P)
                  (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft 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)
                  (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.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) ⋯ ⋯))
                  (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
                  (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)) ⟶
                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))) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) 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) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X ⟶
        TensorBimod.X
          { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
            actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
          P)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X ⟶
      TensorBimod.X
        { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
        P)
    (colimMap
            (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
              (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom)
              (M.X ◁ f.hom) ⋯ ⋯) ▷
          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 :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.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)))
    (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 })
        (AssociatorBimod.homAux M N P ≫
            colimMap
              (parallelPairHom (M.actRight ▷ TensorBimod.X N P)
                (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft 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)
                (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.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) ⋯ ⋯))
                (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) ⋯ ⋯))
                ⋯ ⋯) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) 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) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) P.X ⟶
        TensorBimod.X
          { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
            actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
          P)

Tactic:
dsimp [AssociatorBimod.homAux]

Premises used: Bimod.AssociatorBimod.homAux