-- 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)
X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
S : J.Cover X
x : Meq (FD := FD) (CD := CD) P S
I : S.Arrow
e : (J.pullback I.f).obj (unop (op S)) ⟶ ⊤ := homOfLE ⋯
i : (unop (op ((J.pullback I.f).obj (unop (op S))))).shape.L
h :
  Eq (α := CD (P.obj (op i.Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map (Cover.Arrow.map i e).f.op) :
          (a : CD (P.obj (op I.Y))) → CD (P.obj (op i.Y)))
        ((↑x : (I : S.Arrow) → ToType (P.obj (op I.Y))) I) :
      CD (P.obj (op i.Y)))
    (((@ConcreteCategory.hom.{t, w', w, u_1} _ inst_1 FD CD _ _ _ _
              (Multiequalizer.ι (S.index P) (Cover.Arrow.base i)) :
            FD (multiequalizer (S.index P)) (P.obj (op i.Y))) :
          (a : CD (multiequalizer (S.index P))) → CD (P.obj (op i.Y)))
        ((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 (P.obj (op i.Y)))
⊢ Eq (α := CD (P.obj (op i.Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((S.pullback I.f).index P) i) :
          (a : CD (multiequalizer ((S.pullback I.f).index P))) → CD (P.obj (op i.Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD)
                (Multiequalizer.lift ((S.pullback I.f).index P)
                  (multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
                  (fun (I_1 : (S.pullback I.f).Arrow) ↦ Multiequalizer.ι (Cover.index ⊤ P) (I_1.map e)) ⋯) :
              (a : CD (multiequalizer (⊤.index P))) → CD (multiequalizer ((S.pullback I.f).index P)))
            ((ConcreteCategory.hom (FC := FD) (CC := CD)
                    (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op I.Y))
                      (fun (I_1 : Cover.Arrow.{v, u} (C := C) (X := I.Y) (J := J) ⊤) ↦ P.map I_1.f.op) ⋯) :
                  (a : CD (P.obj (op I.Y))) → CD (multiequalizer (⊤.index P)))
                ((↑x : (I : S.Arrow) → ToType (P.obj (op I.Y))) I) :
              CD (multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))) :
          CD (multiequalizer ((S.pullback I.f).index P))) :
      CD (P.obj (op i.Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι ((S.pullback I.f).index P) i) :
          (a : CD (multiequalizer ((S.pullback I.f).index P))) → CD (P.obj (op i.Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD)
                (Multiequalizer.lift ((S.pullback I.f).index P) (multiequalizer (S.index P))
                  (fun (I_1 : (S.pullback I.f).Arrow) ↦ Multiequalizer.ι (S.index P) I_1.base) ⋯) :
              (a : CD (multiequalizer (S.index P))) → CD (multiequalizer ((S.pullback I.f).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 ((S.pullback I.f).index P))) :
      CD (P.obj (op i.Y)))

Tactic:
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply,
    Multiequalizer.lift_ι, Multiequalizer.lift_ι, Multiequalizer.lift_ι]

Premises used: CategoryTheory.ConcreteCategory.comp_apply, CategoryTheory.Limits.Multiequalizer.lift_ι