Module: Mathlib.CategoryTheory.Monoidal.Free.Coherence

Goal state before:
pp: case id
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ 𝟙 X✝ ≫ (normalizeIsoApp' C X✝ n).hom = (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case α_hom
C : Type u
X Y X✝ Y✝ Z✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (α_ X✝ Y✝ Z✝).hom ≫ (normalizeIsoApp' C (X✝ ⊗ Y✝ ⊗ Z✝) n).hom =
      (normalizeIsoApp' C ((X✝ ⊗ Y✝) ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case α_inv
C : Type u
X Y X✝ Y✝ Z✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (α_ X✝ Y✝ Z✝).inv ≫ (normalizeIsoApp' C ((X✝ ⊗ Y✝) ⊗ Z✝) n).hom =
      (normalizeIsoApp' C (X✝ ⊗ Y✝ ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case l_hom
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (λ_ X✝).hom ≫ (normalizeIsoApp' C X✝ n).hom =
      (normalizeIsoApp' C (𝟙_ (F C) ⊗ X✝) n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case l_inv
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (λ_ X✝).inv ≫ (normalizeIsoApp' C (𝟙_ (F C) ⊗ X✝) n).hom =
      (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case ρ_hom
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (ρ_ X✝).hom ≫ (normalizeIsoApp' C X✝ n).hom =
      (normalizeIsoApp' C (X✝ ⊗ 𝟙_ (F C)) n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case ρ_inv
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (ρ_ X✝).inv ≫ (normalizeIsoApp' C (X✝ ⊗ 𝟙_ (F C)) n).hom =
      (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case whiskerLeft
C : Type u
X Y X✝ Y✝ Z✝ : F C
f✝ : Y✝ ⟶ Z✝
a✝ :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f✝ ≫ (normalizeIsoApp' C Z✝ n).hom = (normalizeIsoApp' C Y✝ n).hom ≫ inclusion.map (eqToHom ⋯)
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ X✝ ◁ f✝ ≫ (normalizeIsoApp' C (X✝ ⊗ Z✝) n).hom =
      (normalizeIsoApp' C (X✝ ⊗ Y✝) n).hom ≫ inclusion.map (eqToHom ⋯)

pp: case whiskerRight
C : Type u
X Y X✝ Y✝ : F C
f✝ : X✝ ⟶ Y✝
Z✝ : F C
a✝ :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f✝ ≫ (normalizeIsoApp' C Y✝ n).hom = (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f✝ ▷ Z✝ ≫ (normalizeIsoApp' C (Y✝ ⊗ Z✝) n).hom =
      (normalizeIsoApp' C (X✝ ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)

Tactic:
case whiskerLeft X' X Y f ih =>
    intro n
    dsimp only [normalizeObj_tensor, normalizeIsoApp'_tensor, Iso.trans_hom,
      Iso.symm_hom, whiskerRightIso_hom, Function.comp_apply, inclusion_obj]
    rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
    simp

Goal state after:
case id
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ 𝟙 X✝ ≫ (normalizeIsoApp' C X✝ n).hom = (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

case α_hom
C : Type u
X Y X✝ Y✝ Z✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (α_ X✝ Y✝ Z✝).hom ≫ (normalizeIsoApp' C (X✝ ⊗ Y✝ ⊗ Z✝) n).hom =
      (normalizeIsoApp' C ((X✝ ⊗ Y✝) ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)

case α_inv
C : Type u
X Y X✝ Y✝ Z✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (α_ X✝ Y✝ Z✝).inv ≫ (normalizeIsoApp' C ((X✝ ⊗ Y✝) ⊗ Z✝) n).hom =
      (normalizeIsoApp' C (X✝ ⊗ Y✝ ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)

case l_hom
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (λ_ X✝).hom ≫ (normalizeIsoApp' C X✝ n).hom =
      (normalizeIsoApp' C (𝟙_ (F C) ⊗ X✝) n).hom ≫ inclusion.map (eqToHom ⋯)

case l_inv
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (λ_ X✝).inv ≫ (normalizeIsoApp' C (𝟙_ (F C) ⊗ X✝) n).hom =
      (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

case ρ_hom
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (ρ_ X✝).hom ≫ (normalizeIsoApp' C X✝ n).hom =
      (normalizeIsoApp' C (X✝ ⊗ 𝟙_ (F C)) n).hom ≫ inclusion.map (eqToHom ⋯)

case ρ_inv
C : Type u
X Y X✝ : F C
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ (ρ_ X✝).inv ≫ (normalizeIsoApp' C (X✝ ⊗ 𝟙_ (F C)) n).hom =
      (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)

case whiskerRight
C : Type u
X Y X✝ Y✝ : F C
f✝ : X✝ ⟶ Y✝
Z✝ : F C
a✝ :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f✝ ≫ (normalizeIsoApp' C Y✝ n).hom = (normalizeIsoApp' C X✝ n).hom ≫ inclusion.map (eqToHom ⋯)
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f✝ ▷ Z✝ ≫ (normalizeIsoApp' C (Y✝ ⊗ Z✝) n).hom =
      (normalizeIsoApp' C (X✝ ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)