-- Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) ⋯)
X B : C
π : X ⟶ B
mpr :
  ∀ [EffectiveEpi π] [inst_2 : HasPullback π π],
    Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
      (fun (X : Type u_4) ↦ X)
        (↑{x : P.obj (op X) |
              Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x :
                  (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x :
                  (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
          Type u_4))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) :
        (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
          (fun (X : Type u_4) ↦ X)
            (↑{x : P.obj (op X) |
                  (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                        (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                          (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                      x =
                    (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                        (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                          (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                      x} :
              Type u_4))
⊢ ∀ [EffectiveEpi π] [inst_2 : HasPullback π π],
    Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
      (fun (X : Type u_4) ↦ X)
        (↑{x : P.obj (op X) |
              Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x :
                  (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x :
                  (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
          Type u_4))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) :
        (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
          (fun (X : Type u_4) ↦ X)
            (↑{x : P.obj (op X) |
                  (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                        (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                          (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                      x =
                    (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                        (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                          (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                      x} :
              Type u_4))

Tactic:
_