-- Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F G H : OplaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
_ :
  η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ θ.naturality (f ≫ g) ⊗≫
        (η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)
_₁ :
  𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ θ.naturality (f ≫ g) ⊗≫
        (η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
        (η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)
_₂ :
  𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
        (η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)
_₃ :
  𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv
⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv

Tactic:
calc
    _ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality (f ≫ g) ⊗≫
          (η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 _ := by
      bicategory
    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
          (η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 _ := by
      rw [η.naturality_comp]
      bicategory
    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
          ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
            F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 _ := by
      rw [θ.naturality_comp]
      bicategory
    _ = _ := by
      rw [whisker_exchange]
      bicategory

Premises used: CategoryTheory.OplaxFunctor.mapComp, CategoryTheory.CategoryStruct.id, CategoryTheory.OplaxFunctor.toPrelaxFunctor, CategoryTheory.Bicategory.whiskerRight, CategoryTheory.PrelaxFunctor.toPrelaxFunctorStruct, CategoryTheory.Oplax.LaxTrans.app, CategoryTheory.Oplax.LaxTrans.naturality, Prefunctor.map, CategoryTheory.CategoryStruct.comp, CategoryTheory.PrelaxFunctorStruct.toPrefunctor, CategoryTheory.Bicategory.whiskerLeft, CategoryTheory.bicategoricalComp