-- 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 : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
erw :
  Eq (α := F.obj (obj (J.diagram P (unop X)) W) ⟶ (J.plusObj (P ⋙ F)).obj X)
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
        (NatTrans.app (Cocone.ι (colimit.cocone (J.diagram P (unop X) ⋙ F))) W :
          obj (J.diagram P (unop X) ⋙ F) W ⟶
            obj
              (obj (const (J.Cover (unop X))ᵒᵖ)
                (Cocone.pt (J := (J.Cover (unop X))ᵒᵖ) (F := J.diagram P (unop X) ⋙ F)
                  (colimit.cocone (J.diagram P (unop X) ⋙ F))))
              W)
        (Iso.hom
            (HasColimit.isoOfNatIso
              (NatIso.ofComponents
                (fun (W : (J.Cover (unop X))ᵒᵖ) ↦
                  Iso.trans (Y := limit (((unop W).index P).multicospan ⋙ F))
                    (IsLimit.conePointUniqueUpToIso
                      (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
                      (limit.isLimit (((unop W).index P).multicospan ⋙ F)))
                    (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm))
                ⋯)) :
          colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F) ⟶ (J.plusObj (P ⋙ F)).obj X) :
      F.obj (obj (J.diagram P (unop X)) W) ⟶ (J.plusObj (P ⋙ F)).obj X)
    (CategoryStruct.comp (Y := obj (J.diagram (P ⋙ F) (unop X)) W)
        (CategoryStruct.comp (Y := limit (((unop W).index P).multicospan ⋙ F))
          (Iso.hom
            (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
              (limit.isLimit (((unop W).index P).multicospan ⋙ F))))
          (Iso.hom (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)))
        (colimit.ι (J.diagram (P ⋙ F) (unop X)) W) :
      obj (J.diagram P (unop X) ⋙ F) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))
⊢ Eq (α := F.obj (obj (J.diagram P (unop X)) W) ⟶ (J.plusObj (P ⋙ F)).obj X)
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X) ⋙ F))
        (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))
          (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
            (NatIso.ofComponents
              (fun (W : (J.Cover (unop X))ᵒᵖ) ↦
                Iso.trans (Y := limit (((unop W).index P).multicospan ⋙ F))
                  (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
                    (limit.isLimit (((unop W).index P).multicospan ⋙ F)))
                  (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm))
              ⋯))) :
      F.obj (obj (J.diagram P (unop X)) W) ⟶ (J.plusObj (P ⋙ F)).obj X)
    (CategoryStruct.comp (Y := obj (J.diagram (P ⋙ F) (unop X)) W)
        (CategoryStruct.comp (Y := limit (((unop W).index P).multicospan ⋙ F))
          (Iso.hom
            (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan))
              (limit.isLimit (((unop W).index P).multicospan ⋙ F))))
          (Iso.hom (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)))
        (colimit.ι (J.diagram (P ⋙ F) (unop X)) W) :
      obj (J.diagram P (unop X) ⋙ F) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (P ⋙ F) (unop X)))

Tactic:
erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).fac]

Premises used: CategoryTheory.Limits.IsColimit.fac, CategoryTheory.Limits.colimit.isColimit, Opposite.unop, CategoryTheory.Limits.isColimitOfPreserves, CategoryTheory.GrothendieckTopology.diagram