-- 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
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
P Q : Functor.{v, v_1, u, u_1} Cᵒᵖ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
w :
  Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
    (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 X))ᵒᵖ) (J.diagram Q (unop X))))
            (F.map (colimMap (J.diagramNatTrans η (unop X))) :
              F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) ⟶
                F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram Q (unop X))))
            (NatTrans.app (Iso.hom (J.plusCompIso F Q)) X :
              obj.{v, v_2, u, u_2}
                  ({ obj := fun (X : Cᵒᵖ) ↦ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram Q (unop X)),
                      map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) ↦
                        CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram Q (unop Y)))
                          (colimMap (J.diagramPullback Q f.unop))
                          (colimit.pre (J.diagram Q (unop Y)) (J.pullback f.unop).op),
                      map_id := ⋯, map_comp := ⋯ : Functor.{v, v_1, u, u_1} Cᵒᵖ D } ⋙
                    F)
                  X ⟶
                obj.{v, v_2, u, u_2}
                  { obj := fun (X : Cᵒᵖ) ↦ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)),
                    map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) ↦
                      CategoryStruct.comp (Y := colimit ((J.pullback f.unop).op ⋙ J.diagram (Q ⋙ F) (unop Y)))
                        (colimMap (J.diagramPullback (Q ⋙ F) f.unop))
                        (colimit.pre (J.diagram (Q ⋙ F) (unop Y)) (J.pullback f.unop).op),
                    map_id := ⋯, map_comp := ⋯ }
                  X) :
          F.obj (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) ⟶
            colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
    (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)))
            (NatTrans.app (Iso.hom (J.plusCompIso F P)) X :
              obj.{v, v_2, u, u_2}
                  ({ obj := fun (X : Cᵒᵖ) ↦ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)),
                      map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) ↦
                        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),
                      map_id := ⋯, map_comp := ⋯ : Functor.{v, v_1, u, u_1} Cᵒᵖ D } ⋙
                    F)
                  X ⟶
                obj.{v, v_2, u, u_2}
                  { obj := fun (X : Cᵒᵖ) ↦ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)),
                    map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) ↦
                      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),
                    map_id := ⋯, map_comp := ⋯ }
                  X)
            (colimMap (J.diagramNatTrans (whiskerRight η F) (unop X)) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)) ⟶
                colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (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 (Q ⋙ F) (unop X))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
⊢ Eq (α := obj (J.diagram P (unop X) ⋙ F) W ⟶ (J.plusObj (Q ⋙ F)).obj X)
    (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)
      (NatTrans.app (whiskerRight (J.plusMap η) F ≫ Iso.hom (J.plusCompIso F Q)) X))
    (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)
      (NatTrans.app
        (CategoryStruct.comp (Y := J.plusObj (P ⋙ F))
          (Iso.hom (J.plusCompIso F P) : J.plusObj P ⋙ F ⟶ J.plusObj (P ⋙ F))
          (J.plusMap (whiskerRight η F) : J.plusObj (P ⋙ F) ⟶ J.plusObj (Q ⋙ F)))
        X))

Tactic:
dsimp [plusObj, plusMap]

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