-- 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 : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
h :
  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)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) Y.X ◁ f.hom) (TensorBimod.X M N ◁ f.hom) ⋯ ⋯)) =
    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 := ⋯, 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) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ f.hom)
                      (N.X ◁ f.hom) ⋯ ⋯))
                (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) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ f.hom)
                      (N.X ◁ f.hom) ⋯ ⋯))
                ⋯ ⋯) :
            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')
⊢ 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
        ({ 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 }.whiskerLeft
          f) =
    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 (N.whiskerLeft f) ≫
            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 } }
                ⋯ ⋯))

Tactic:
dsimp