-- 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 M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
h :
  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 } }
                ⋯ ⋯))
⊢ 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 := ⋯ }) =
    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 } }
              ⋯ ⋯))

Tactic:
apply coequalizer.hom_ext

Premises used: CategoryTheory.Limits.coequalizer.hom_ext