Theorem: CategoryTheory.GlueData.ι_gluedIso_hom
File: Mathlib/CategoryTheory/GlueData.lean

tactic: haveI : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F))) := inferInstance
state_before: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
⊢ F.map (D.ι i) ≫ (D.gluedIso F).hom = (D.mapGlueData F).ι i
state_after: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ F.map (D.ι i) ≫ (D.gluedIso F).hom = (D.mapGlueData F).ι i

tactic: erw [ι_preservesColimitsIso_hom_assoc]
state_before: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ F.map (D.ι i) ≫ (D.gluedIso F).hom = (D.mapGlueData F).ι i
state_after: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ colimit.ι (D.diagram.multispan ⋙ F) (WalkingMultispan.right i) ≫ (HasColimit.isoOfNatIso (D.diagramIso F)).hom =
    (D.mapGlueData F).ι i

tactic: rw [HasColimit.isoOfNatIso_ι_hom]
state_before: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ colimit.ι (D.diagram.multispan ⋙ F) (WalkingMultispan.right i) ≫ (HasColimit.isoOfNatIso (D.diagramIso F)).hom =
    (D.mapGlueData F).ι i
state_after: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ (D.diagramIso F).hom.app (WalkingMultispan.right i) ≫
      colimit.ι (D.mapGlueData F).diagram.multispan (WalkingMultispan.right i) =
    (D.mapGlueData F).ι i

tactic: erw [Category.id_comp]
state_before: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ (D.diagramIso F).hom.app (WalkingMultispan.right i) ≫
      colimit.ι (D.mapGlueData F).diagram.multispan (WalkingMultispan.right i) =
    (D.mapGlueData F).ι i
state_after: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ colimit.ι (D.mapGlueData F).diagram.multispan (WalkingMultispan.right i) = (D.mapGlueData F).ι i

tactic: rfl
state_before: C : Type u₁
inst✝³ : Category.{v, u₁} C
C' : Type u₂
inst✝² : Category.{v, u₂} C'
D : GlueData C
F : C ⥤ C'
H : (i j k : D.J) → PreservesLimit (cospan (D.f i j) (D.f i k)) F
inst✝¹ : HasMulticoequalizer D.diagram
inst✝ : PreservesColimit D.diagram.multispan F
i : D.J
this : HasColimit (D.mapGlueData F).diagram.multispan
⊢ colimit.ι (D.mapGlueData F).diagram.multispan (WalkingMultispan.right i) = (D.mapGlueData F).ι i
state_after: no goals