-- Module: Mathlib.CategoryTheory.Sites.CompatiblePlus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
P : Functor.{v, v_1, u, u_1} Cᵒᵖ D
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X Y : Cᵒᵖ
f : X ⟶ Y
W : (J.Cover (unop X))ᵒᵖ
dsimp :
  Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
        (F.map (colimit.ι (J.diagram P (unop X)) W) :
          F.obj (multiequalizer ((unop W).index P)) ⟶
            F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
        (CategoryStruct.comp (Y := F.obj (colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y))))
            (F.map
              (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)))
                (colimMap (J.diagramPullback P f.unop)) (colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op)))
            (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y) ⋙ F))
              (Iso.hom
                (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop Y))))
                  (colimit.isColimit (J.diagram P (unop Y) ⋙ F))))
              (Iso.hom (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop Y))))) :
          F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) ⟶
            colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
    (CategoryStruct.comp (Y := F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
        (F.map (colimit.ι (J.diagram P (unop X)) W) :
          F.obj (multiequalizer ((unop W).index P)) ⟶
            F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))))
        (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))
            (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
                (Iso.hom
                  (IsColimit.coconePointUniqueUpToIso
                    (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X))))
                    (colimit.isColimit (J.diagram P (unop X) ⋙ F))))
                (Iso.hom (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop X)))) :
              F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) ⟶
                colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))
            (CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram (P ⋙ F) (unop Y)))
                (colimMap (J.diagramPullback (P ⋙ F) f.unop))
                (colimit.pre (J.diagram (P ⋙ F) (unop Y)) (J.pullback f.unop).op) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)) ⟶
                colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
          F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) ⟶
            colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram (P ⋙ F) (unop Y)))
⊢ Eq (α := obj (J.diagram P (unop X) ⋙ F) W ⟶ (J.plusObj (P ⋙ F)).obj Y)
    (CategoryStruct.comp (Y :=
      obj
        (obj (const (J.Cover (unop X))ᵒᵖ)
          (Cocone.pt (J := (J.Cover (unop X))ᵒᵖ) (F := J.diagram P (unop X) ⋙ F)
            (F.mapCocone (colimit.cocone (J.diagram P (unop X))))))
        W)
      (NatTrans.app (Cocone.ι (F.mapCocone (colimit.cocone (J.diagram P (unop X))))) W)
      ((J.plusObj P ⋙ F).map f ≫
        Iso.hom
          ((fun (X : Cᵒᵖ) ↦
              (Iso.trans (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
                  (IsColimit.coconePointUniqueUpToIso
                    (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X))))
                    (colimit.isColimit (J.diagram P (unop X) ⋙ F)))
                  (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop X))) :
                (J.plusObj P ⋙ F).obj X ≅ (J.plusObj (P ⋙ F)).obj X))
            Y)))
    (CategoryStruct.comp (Y :=
      obj
        (obj (const (J.Cover (unop X))ᵒᵖ)
          (Cocone.pt (J := (J.Cover (unop X))ᵒᵖ) (F := J.diagram P (unop X) ⋙ F)
            (F.mapCocone (colimit.cocone (J.diagram P (unop X))))))
        W)
      (NatTrans.app (Cocone.ι (F.mapCocone (colimit.cocone (J.diagram P (unop X))))) W)
      (Iso.hom
          ((fun (X : Cᵒᵖ) ↦
              (Iso.trans (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
                  (IsColimit.coconePointUniqueUpToIso
                    (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X))))
                    (colimit.isColimit (J.diagram P (unop X) ⋙ F)))
                  (HasColimit.isoOfNatIso (J.diagramCompIso F P (unop X))) :
                (J.plusObj P ⋙ F).obj X ≅ (J.plusObj (P ⋙ F)).obj X))
            X) ≫
        (J.plusObj (P ⋙ F)).map f))

Tactic:
dsimp [plusObj, plusMap]

Premises used: CategoryTheory.GrothendieckTopology.plusMap, CategoryTheory.GrothendieckTopology.plusObj