-- Module: Mathlib.CategoryTheory.Sites.ConcreteSheafification
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
S T : J.Cover X
x : Meq (FD := FD) (CD := CD) P S
y : Meq (FD := FD) (CD := CD) P T
h :
  Eq (α := ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))) (mk (FD := FD) (CD := CD) x)
    (mk (FD := FD) (CD := CD) y)
W : (J.Cover X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : (unop W).Arrow
hh :
  Eq.{t + 1} (α := CD (((unop W).index P).left I))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((unop W).index P) I) :
          (a : CD (multiequalizer ((unop W).index P))) → CD (((unop W).index P).left I))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) ((J.diagram P X).map h1) :
              (a : CD ((J.diagram P X).obj (op S))) → CD ((J.diagram P X).obj W))
            ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P S) : (a : Meq P S) → ToType (multiequalizer (S.index P)))
                x :
              ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer (S.index P))) :
          CD ((J.diagram P X).obj W)) :
      CD (((unop W).index P).left I))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((unop W).index P) I) :
          (a : CD (multiequalizer ((unop W).index P))) → CD (((unop W).index P).left I))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) ((J.diagram P X).map h2) :
              (a : CD ((J.diagram P X).obj (op T))) → CD ((J.diagram P X).obj W))
            ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P T) : (a : Meq P T) → ToType (multiequalizer (T.index P)))
                y :
              ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer (T.index P))) :
          CD ((J.diagram P X).obj W)) :
      CD (((unop W).index P).left I))
e_1 : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (P.obj (op I.Y)) = CD (((unop W).index P).left I)
h₁ :
  (↑x : (I : S.Arrow) → ToType (P.obj (op I.Y))) { Y := I.Y, f := I.f, hf := ⋯ : S.Arrow } =
    (ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((unop W).index P) I) :
        (a : CD (multiequalizer ((unop W).index P))) → CD (P.obj (op I.Y)))
      ((ConcreteCategory.hom (FC := FD) (CC := CD)
              (Multiequalizer.lift ((unop W).index P) (multiequalizer (S.index P))
                (fun (I : (unop W).Arrow) ↦ Multiequalizer.ι (S.index P) (I.map h1.unop)) ⋯) :
            (a : CD (multiequalizer (S.index P))) → CD (multiequalizer ((unop W).index P)))
          ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P S) : (a : Meq P S) → ToType (multiequalizer (S.index P))) x :
            ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer (S.index P))) :
        CD (multiequalizer ((unop W).index P)))
⊢ (↑(x.refine (FD := FD) (CD := CD) h1.unop) : (I : (unop W).Arrow) → ToType (P.obj (op I.Y))) I =
    (ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((unop W).index P) I) :
        (a : CD (multiequalizer ((unop W).index P))) → CD (((unop W).index P).left I))
      ((ConcreteCategory.hom (FC := FD) (CC := CD) ((J.diagram P X).map h1) :
            (a : CD ((J.diagram P X).obj (op S))) → CD ((J.diagram P X).obj W))
          ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P S) : (a : Meq P S) → ToType (multiequalizer (S.index P))) x :
            ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer (S.index P))) :
        CD ((J.diagram P X).obj W))

Tactic:
dsimp [diagram]

Premises used: CategoryTheory.GrothendieckTopology.diagram