-- Module: Mathlib.CategoryTheory.Sites.EqualizerSheafCondition
Goal:
C : Type u
inst : Category.{v, u} C
P : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
B : C
I : Type t
inst_1 : Small.{w, t} I
X : I → C
π : (i : I) → X i ⟶ B
inst_2 : (Presieve.ofArrows X π).HasPairwisePullbacks
x : FirstObj P X
i j : I
t :
  Eq (α :=
    (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (X := X (i, j).1) (Y := X (i, j).2) (π (i, j).1) (π (i, j).2)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (Pi.π (fun (ij : I × I) ↦ P.obj (op (Limits.pullback (X := X ij.1) (Y := X ij.2) (π ij.1) (π ij.2))))
              (i, j)) :
          (a : (fun (X : Type w) ↦ X) (∏ᶜ fun (ij : I × I) ↦ P.obj (op (Limits.pullback (π ij.1) (π ij.2))))) →
            (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π (i, j).1) (π (i, j).2)))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (firstMap P X π) :
              (a : (fun (X : Type w) ↦ X) (FirstObj P X)) → (fun (X : Type w) ↦ X) (SecondObj P X π))
            x :
          (fun (X : Type w) ↦ X) (SecondObj P X π)) :
      (fun (X : Type w) ↦ X)
        (P.obj (op (Limits.pullback (X := X (i, j).1) (Y := X (i, j).2) (π (i, j).1) (π (i, j).2)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (Pi.π (fun (ij : I × I) ↦ P.obj (op (Limits.pullback (X := X ij.1) (Y := X ij.2) (π ij.1) (π ij.2))))
              (i, j)) :
          (a : (fun (X : Type w) ↦ X) (∏ᶜ fun (ij : I × I) ↦ P.obj (op (Limits.pullback (π ij.1) (π ij.2))))) →
            (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π (i, j).1) (π (i, j).2)))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (secondMap P X π) :
              (a : (fun (X : Type w) ↦ X) (FirstObj P X)) → (fun (X : Type w) ↦ X) (SecondObj P X π))
            x :
          (fun (X : Type w) ↦ X) (SecondObj P X π)) :
      (fun (X : Type w) ↦ X)
        (P.obj (op (Limits.pullback (X := X (i, j).1) (Y := X (i, j).2) (π (i, j).1) (π (i, j).2)))))
⊢ Eq (α := (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π i) (π j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst (π i) (π j)))) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op (X i)))) →
            (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π i) (π j)))))
        ((Equiv.symm (equivShrink ((i : I) → P.obj (op (X i)))) :
              (a : Shrink.{w, max t w} ((i : I) → P.obj (op (X i)))) → (i : I) → P.obj (op (X i)))
            (((ConcreteCategory.hom (FC := TypeCat.Fun)
                      (Iso.hom (Types.Small.productIso fun (i : I) ↦ P.obj (op (X i)))) :
                    TypeCat.Fun (∏ᶜ fun (i : I) ↦ P.obj (op (X i)))
                      (Shrink.{w, max w t} ((j : I) → P.obj (op (X j))))) :
                  (a : (fun (X : Type w) ↦ X) (∏ᶜ fun (i : I) ↦ P.obj (op (X i)))) →
                    (fun (X : Type w) ↦ X) (Shrink.{w, max w t} ((j : I) → P.obj (op (X j)))))
                x :
              (fun (X : Type w) ↦ X) (Shrink.{w, max w t} ((j : I) → P.obj (op (X j)))))
            i :
          P.obj (op (X i))) :
      (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π i) (π j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd (π i) (π j)))) :
          (a : (fun (X : Type w) ↦ X) (P.obj (op (X j)))) →
            (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π i) (π j)))))
        ((Equiv.symm (equivShrink ((i : I) → P.obj (op (X i)))) :
              (a : Shrink.{w, max t w} ((i : I) → P.obj (op (X i)))) → (i : I) → P.obj (op (X i)))
            (((ConcreteCategory.hom (FC := TypeCat.Fun)
                      (Iso.hom (Types.Small.productIso fun (i : I) ↦ P.obj (op (X i)))) :
                    TypeCat.Fun (∏ᶜ fun (i : I) ↦ P.obj (op (X i)))
                      (Shrink.{w, max w t} ((j : I) → P.obj (op (X j))))) :
                  (a : (fun (X : Type w) ↦ X) (∏ᶜ fun (i : I) ↦ P.obj (op (X i)))) →
                    (fun (X : Type w) ↦ X) (Shrink.{w, max w t} ((j : I) → P.obj (op (X j)))))
                x :
              (fun (X : Type w) ↦ X) (Shrink.{w, max w t} ((j : I) → P.obj (op (X j)))))
            j :
          P.obj (op (X j))) :
      (fun (X : Type w) ↦ X) (P.obj (op (Limits.pullback (π i) (π j)))))

Tactic:
simpa [firstMap, secondMap] using t

Premises used: CategoryTheory.Equalizer.Presieve.Arrows.firstMap, CategoryTheory.Equalizer.Presieve.Arrows.secondMap