-- 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 : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
⊢ 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 := ⋯ }.whiskerLeft
        f) =
    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 } }
              ⋯ ⋯))
have h : coequalizer.π
      ({ 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 }.actRight ▷
        P.X)
      (Iso.hom
          (α_
            { 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 }.X
            Y.X P.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 }.X ◁
          P.actLeft) ≫
    Hom.hom
      ({ 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 }.whiskerLeft
        f) =
  coequalizer.π
      ({ 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 }.actRight ▷
        P.X)
      (Iso.hom
          (α_
            { 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 }.X
            Y.X P.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 }.X ◁
          P.actLeft) ≫
    Hom.hom
      (Iso.hom
          (isoOfIso
            { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := sorry, inv_hom_id := sorry :
              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 ≅
                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 } }
            sorry sorry) ≫
        M.whiskerLeft (N.whiskerLeft f) ≫
          Iso.inv
            (isoOfIso
              { hom := AssociatorBimod.hom M N P', inv := AssociatorBimod.inv M N P', hom_inv_id := sorry, inv_hom_id := sorry :
                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' ≅
                  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 } }
              sorry sorry))
Tactic: apply coequalizer.hom_ext