-- Module: Mathlib.AlgebraicGeometry.Cover.Open
Goal:
X Y Z : Scheme
𝒰_1 : X.OpenCover
f_1 : X ⟶ Z
g : Y ⟶ Z
inst : ∀ (x : 𝒰_1.I₀), HasPullback (𝒰_1.f x ≫ f_1) g
f : X ⟶ Y
𝒰 : OpenCover.{u_1, u} Y
i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).I₀
⊢ Eq (α :=
    pullback (X :=
        𝒰.X
          (i.fst (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))))
        (𝒰.f
          (i.fst (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))))
        f ⟶
      𝒰.X
        (i.fst (β := fun (i : 𝒰.I₀) ↦
          PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
            ((fun (i : 𝒰.I₀) ↦
                (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                    ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                  PreZeroHypercover (𝒰.X i)))
              i))))
    (pullback.fst
        (𝒰.f
          (i.fst (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))))
        f ≫
      𝟙
        (𝒰.X
          (i.fst (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i)))))
    (Iso.hom
        (pullbackSymmetry
          (𝒰.f
            (i.fst (β := fun (i : 𝒰.I₀) ↦
              PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
                ((fun (i : 𝒰.I₀) ↦
                    (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                        ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                      PreZeroHypercover (𝒰.X i)))
                  i))))
          f) ≫
      Cover.pullbackHom (P := IsOpenImmersion) 𝒰 f
        (i.fst (β := fun (i : 𝒰.I₀) ↦
          PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
            ((fun (i : 𝒰.I₀) ↦
                (Precoverage.ZeroHypercover.toPreZeroHypercover (J := precoverage IsOpenImmersion)
                    ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                  PreZeroHypercover (𝒰.X i)))
              i))))

Tactic:
simp [Cover.pullbackHom]

Premises used: AlgebraicGeometry.Scheme.Cover.pullbackHom