-- 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)
Y X : C
P : Functor.{v, w', u, w} Cᵒᵖ D
S : J.Cover X
x : Meq (FD := FD) (CD := CD) P S
f : Y ⟶ X
rw :
  Eq.{t + 1} (α := CD (colimit (J.diagram P Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (colimit.ι (J.diagram P Y) ((J.pullback f).op.obj (op S))) :
          (a : CD (((J.pullback f).op ⋙ J.diagram P Y).obj (op S))) → CD (colimit (J.diagram P Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) (NatTrans.app (J.diagramPullback P f) (op S)) :
              (a : CD (multiequalizer (S.index P))) → CD (((J.pullback f).op ⋙ J.diagram P Y).obj (op S)))
            ((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.pullback f).op ⋙ J.diagram P Y).obj (op S))) :
      CD (colimit (J.diagram P Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (colimit.ι (J.diagram P Y) (op (S.pullback f))) :
          (a : CD (multiequalizer ((S.pullback f).index P))) → CD (colimit (J.diagram P Y)))
        ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P (S.pullback f)) :
              (a : Meq P (S.pullback f)) → ToType (multiequalizer ((S.pullback f).index P)))
            (x.pullback (FD := FD) (CD := CD) f) :
          ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer ((S.pullback f).index P))) :
      CD (colimit (J.diagram P Y)))
⊢ Eq.{t + 1} (α := CD (colimit (J.diagram P Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD)
            (CategoryStruct.comp (Y := colimit ((J.pullback f).op ⋙ J.diagram P Y)) (colimMap (J.diagramPullback P f))
              (colimit.pre (J.diagram P Y) (J.pullback f).op)) :
          (a : CD (colimit (J.diagram P X))) → CD (colimit (J.diagram P Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) (colimit.ι (J.diagram P X) (op S)) :
              (a : CD (multiequalizer (S.index P))) → CD (colimit (J.diagram P X)))
            ((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 (colimit (J.diagram P X))) :
      CD (colimit (J.diagram P Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (colimit.ι (J.diagram P Y) (op (S.pullback f))) :
          (a : CD (multiequalizer ((S.pullback f).index P))) → CD (colimit (J.diagram P Y)))
        ((Equiv.symm (Meq.equiv (FD := FD) (CD := CD) P (S.pullback f)) :
              (a : Meq P (S.pullback f)) → ToType (multiequalizer ((S.pullback f).index P)))
            (x.pullback (FD := FD) (CD := CD) f) :
          ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (multiequalizer ((S.pullback f).index P))) :
      CD (colimit (J.diagram P Y)))

Tactic:
rw [← CategoryTheory.comp_apply (x := (Meq.equiv P S).symm x), ι_colimMap_assoc, colimit.ι_pre,
    CategoryTheory.comp_apply (x := (Meq.equiv P S).symm x)]

Premises used: CategoryTheory.Limits.ι_colimMap_assoc, CategoryTheory.Limits.colimit.ι_pre, CategoryTheory.Meq.equiv, CategoryTheory.comp_apply, Equiv.symm, DFunLike.coe