-- 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 :
  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')
    (TensorBimod.X M N ◁ f.hom ≫
        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 :
      (parallelPair (TensorBimod.actRight M N ▷ P.X)
              (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft)).obj
          WalkingParallelPair.one ⟶
        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)))
    (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') :
      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')
⊢ 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')

Tactic:
slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one