Theorem: CategoryTheory.GrothendieckTopology.ι_plusCompIso_hom
File: Mathlib/CategoryTheory/Sites/CompatiblePlus.lean

Proof steps:
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W) ≫ (J.plusCompIso F P).hom.app X =
    (J.diagramCompIso F P (unop X)).hom.app W ≫ colimit.ι (J.diagram (P ⋙ F) (unop X)) W
state_after: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W) ≫
      (NatIso.ofComponents
              (fun X =>
                (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso
                    (colimit.isColimit (J.diagram P (unop X) ⋙ F)) ≪≫
                  HasColimit.isoOfNatIso
                    (NatIso.ofComponents
                      (fun W =>
                        (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                            (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                          HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
                      ⋯))
              ⋯).hom.app
        X =
    (NatIso.ofComponents
              (fun W =>
                (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                    (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                  HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
              ⋯).hom.app
        W ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
tactic: delta diagramCompIso plusCompIso
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W) ≫
      (NatIso.ofComponents
              (fun X =>
                (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso
                    (colimit.isColimit (J.diagram P (unop X) ⋙ F)) ≪≫
                  HasColimit.isoOfNatIso
                    (NatIso.ofComponents
                      (fun W =>
                        (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                            (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                          HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
                      ⋯))
              ⋯).hom.app
        X =
    (NatIso.ofComponents
              (fun W =>
                (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                    (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                  HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
              ⋯).hom.app
        W ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
state_after: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ (F.map (colimit.ι (J.diagram P (unop X)) W) ≫
        ((isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso
            (colimit.isColimit (J.diagram P (unop X) ⋙ F))).hom) ≫
      (HasColimit.isoOfNatIso
          (NatIso.ofComponents
            (fun W =>
              (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
            ⋯)).hom =
    (((isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
            (limit.isLimit (((unop W).index P).multicospan ⋙ F))).hom ≫
        (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm).hom) ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
tactic: simp only [IsColimit.descCoconeMorphism_hom, IsColimit.uniqueUpToIso_hom,
    Cocones.forget_map, Iso.trans_hom, NatIso.ofComponents_hom_app, Functor.mapIso_hom, ←
    Category.assoc]
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ (F.map (colimit.ι (J.diagram P (unop X)) W) ≫
        ((isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso
            (colimit.isColimit (J.diagram P (unop X) ⋙ F))).hom) ≫
      (HasColimit.isoOfNatIso
          (NatIso.ofComponents
            (fun W =>
              (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
            ⋯)).hom =
    (((isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
            (limit.isLimit (((unop W).index P).multicospan ⋙ F))).hom ≫
        (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm).hom) ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
state_after: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ (colimit.cocone (J.diagram P (unop X) ⋙ F)).ι.app W ≫
      (HasColimit.isoOfNatIso
          (NatIso.ofComponents
            (fun W =>
              (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
            ⋯)).hom =
    (((isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
            (limit.isLimit (((unop W).index P).multicospan ⋙ F))).hom ≫
        (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm).hom) ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
tactic: erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).fac]
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ (colimit.cocone (J.diagram P (unop X) ⋙ F)).ι.app W ≫
      (HasColimit.isoOfNatIso
          (NatIso.ofComponents
            (fun W =>
              (isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
                  (limit.isLimit (((unop W).index P).multicospan ⋙ F)) ≪≫
                HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm)
            ⋯)).hom =
    (((isLimitOfPreserves F (limit.isLimit ((unop W).index P).multicospan)).conePointUniqueUpToIso
            (limit.isLimit (((unop W).index P).multicospan ⋙ F))).hom ≫
        (HasLimit.isoOfNatIso (Cover.multicospanComp F P (unop W)).symm).hom) ≫
      colimit.ι (J.diagram (P ⋙ F) (unop X)) W
state_after: no goals
tactic: simp