-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
⊢ 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 :=
        coequalizer (TensorBimod.actRight M N ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
        (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
        (coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ≫
          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 })
            (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)))
            (AssociatorBimod.inv M N' P)) :
      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)
have 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 := sorry, left_assoc := sorry,
        actRight := TensorBimod.actRight M N', actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry : 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) sorry sorry) ▷
        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 := sorry, left_assoc := sorry,
          actRight := TensorBimod.actRight N' P, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry : 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) sorry sorry))
              (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) sorry sorry))
              sorry sorry) :
        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 := sorry, left_assoc := sorry,
              actRight := TensorBimod.actRight N' P, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry :
              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 := sorry, left_assoc := sorry,
          actRight := TensorBimod.actRight M N', actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry : Bimod W Y }
        P)
Tactic: slice_rhs 1 2 => rw [coequalizer.π_desc]