-- 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 M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ coequalizer.π
        (M.actRight ▷
          { 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 }.X)
        (Iso.hom
            (α_ M.X X.X
              { 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 }.X) ≫
          M.X ◁
            { 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 := ⋯ }.actLeft) ≫
      Hom.hom
        (whiskerRight f
          { 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 }) =
    coequalizer.π
        (M.actRight ▷
          { 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 }.X)
        (Iso.hom
            (α_ M.X X.X
              { 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 }.X) ≫
          M.X ◁
            { 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 }.actLeft) ≫
      Hom.hom
        (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 } }
              ⋯ ⋯) ≫
          whiskerRight (whiskerRight f N) P ≫
            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 } }
                ⋯ ⋯))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X (TensorBimod.X N P) ⟶
    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 })
  (CategoryStruct.comp (Y :=
    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))
    (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))
    (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) (f.hom ▷ X.X ▷ TensorBimod.X N P)
        (f.hom ▷ TensorBimod.X N P) sorry sorry)))
  (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) ≫
    AssociatorBimod.inv M N P ≫
      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) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) 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) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) sorry sorry) ▷
              P.X)
            sorry sorry) ≫
        AssociatorBimod.hom M' N P)
Tactic: dsimp