-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
X Y : D
f : X ⟶ Y
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) X (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
      ((f ⊗ₘ 𝟙 (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))) ≫ Iso.hom (ρ_ Y)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (Iso.hom (ρ_ X) ≫ f))
have a : Eq (α :=
  Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
      (tensorObj.{v₃, u₃} (𝒞 := inst_8) X (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y))
  (CategoryStruct.comp (Y :=
      Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) Y (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
      (map (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f)
        (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_8)))))
      (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (Iso.hom (ρ_ Y))) :
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) X (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y))
  (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (Iso.hom (ρ_ X)) ≫
      Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) f :
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) X (tensorUnit.{v₃, u₃} D (𝒞 := inst_8))))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) Y))
Tactic: simp only [Functor.map_comp, ι_map_tensorHom_hom_eq_tensorHom, Functor.map_id]