Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
x✝ : Arrow C₂
⊢ Arrow.Hom.left
      ((F.leibnizPushout.obj X₁).map
          ({
                app := fun X₂ ↦
                  Arrow.homMk (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                    (pullback.lift
                      (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                      (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯)
                    ⋯,
                naturality := ⋯ }.app
            x✝) ≫
        {
              app := fun X₃ ↦
                Arrow.homMk
                  (pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
                    (adj₂.homEquiv.symm
                      (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                        ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                    ⋯)
                  (adj₂.homEquiv.symm
                    (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                      ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                  ⋯,
              naturality := ⋯ }.app
          ((F.leibnizPushout.obj X₁).obj x✝)) =
    Arrow.Hom.left (𝟙 ((F.leibnizPushout.obj X₁).obj x✝))

Tactic:
apply pushout.hom_ext

Goal state after:
case h₁.h₀
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
x✝ : Arrow C₂
⊢ pushout.inl ((F.map X₁.hom).app x✝.left) ((F.obj X₁.left).map x✝.hom) ≫
      Arrow.Hom.left
        ((F.leibnizPushout.obj X₁).map
            ({
                  app := fun X₂ ↦
                    Arrow.homMk (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                      (pullback.lift
                        (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                        (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯)
                      ⋯,
                  naturality := ⋯ }.app
              x✝) ≫
          {
                app := fun X₃ ↦
                  Arrow.homMk
                    (pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
                      (adj₂.homEquiv.symm
                        (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                          ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                      ⋯)
                    (adj₂.homEquiv.symm
                      (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                        ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                    ⋯,
                naturality := ⋯ }.app
            ((F.leibnizPushout.obj X₁).obj x✝)) =
    pushout.inl ((F.map X₁.hom).app x✝.left) ((F.obj X₁.left).map x✝.hom) ≫
      Arrow.Hom.left (𝟙 ((F.leibnizPushout.obj X₁).obj x✝))

case h₁
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
x✝ : Arrow C₂
⊢ pushout.inr ((F.map X₁.hom).app x✝.left) ((F.obj X₁.left).map x✝.hom) ≫
      Arrow.Hom.left
        ((F.leibnizPushout.obj X₁).map
            ({
                  app := fun X₂ ↦
                    Arrow.homMk (adj₂.homEquiv (pushout.inl ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                      (pullback.lift
                        (adj₂.homEquiv (pushout.inr ((F.map X₁.hom).app X₂.left) ((F.obj X₁.left).map X₂.hom)))
                        (adj₂.homEquiv (𝟙 ((F.obj (unop (op X₁)).right).obj ((𝟭 (Arrow C₂)).obj X₂).right))) ⋯)
                      ⋯,
                  naturality := ⋯ }.app
              x✝) ≫
          {
                app := fun X₃ ↦
                  Arrow.homMk
                    (pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
                      (adj₂.homEquiv.symm
                        (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                          ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                      ⋯)
                    (adj₂.homEquiv.symm
                      (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom)
                        ((G.map (unop (op X₁)).hom.op).app X₃.right)))
                    ⋯,
                naturality := ⋯ }.app
            ((F.leibnizPushout.obj X₁).obj x✝)) =
    pushout.inr ((F.map X₁.hom).app x✝.left) ((F.obj X₁.left).map x✝.hom) ≫
      Arrow.Hom.left (𝟙 ((F.leibnizPushout.obj X₁).obj x✝))