-- 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
⊢ coequalizer.π
        ({ 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 := ⋯ }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { 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 }.X
              Y.X P.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 }.X ◁
            P.actLeft) ≫
      Hom.hom (whiskerRight (M.whiskerLeft f) P) =
    coequalizer.π
        ({ 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 }.actRight ▷
          P.X)
        (Iso.hom
            (α_
              { 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 }.X
              Y.X P.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 }.X ◁
            P.actLeft) ≫
      Hom.hom
        (Iso.hom
            (isoOfIso
              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
                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 ≅
                  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.whiskerLeft (whiskerRight f P) ≫
            Iso.inv
              (isoOfIso
                { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯,
                  inv_hom_id := ⋯ :
                  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 ≅
                    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 } }
                ⋯ ⋯))
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)
  (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))
    (colimMap
      (parallelPairHom (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)
        (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) ▷
            Y.X ▷
          P.X)
        (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)
        sorry sorry)))
  (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) ≫
    AssociatorBimod.hom M N P ≫
      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 })
        (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) :
          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))
Tactic: dsimp