[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : PreservesLimit (Functor.empty C) G
F : Discrete PEmpty ⥤ D
⊢ HasLimit F
[PROOFSTEP]
haveI := HasLimit.mk ⟨_, isLimitOfHasTerminalOfPreservesLimit G⟩
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : PreservesLimit (Functor.empty C) G
F : Discrete PEmpty ⥤ D
this : HasLimit (Functor.empty D)
⊢ HasLimit F
[PROOFSTEP]
apply hasLimitOfIso F.uniqueFromEmpty.symm
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : HasTerminal D
i : IsIso (terminalComparison G)
⊢ PreservesLimit (Functor.empty C) G
[PROOFSTEP]
apply preservesLimitOfPreservesLimitCone terminalIsTerminal
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : HasTerminal D
i : IsIso (terminalComparison G)
⊢ IsLimit (G.mapCone (asEmptyCone (⊤_ C)))
[PROOFSTEP]
apply (isLimitMapConeEmptyConeEquiv _ _).symm _
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : HasTerminal D
i : IsIso (terminalComparison G)
⊢ IsTerminal (G.obj (⊤_ C))
[PROOFSTEP]
exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (Functor.empty.{0} D)) i
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : HasTerminal D
f : G.obj (⊤_ C) ⟶ ⊤_ D
i : IsIso f
⊢ PreservesLimit (Functor.empty C) G
[PROOFSTEP]
rw [Subsingleton.elim f (terminalComparison G)] at i 
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasTerminal C
inst✝ : HasTerminal D
f : G.obj (⊤_ C) ⟶ ⊤_ D
i : IsIso (terminalComparison G)
⊢ PreservesLimit (Functor.empty C) G
[PROOFSTEP]
exact PreservesTerminal.ofIsoComparison G
[GOAL]
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝² : HasTerminal C
inst✝¹ : HasTerminal D
inst✝ : PreservesLimit (Functor.empty C) G
⊢ IsIso (terminalComparison G)
[PROOFSTEP]
rw [← PreservesTerminal.iso_hom]
[GOAL]
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝² : HasTerminal C
inst✝¹ : HasTerminal D
inst✝ : PreservesLimit (Functor.empty C) G
⊢ IsIso (PreservesTerminal.iso G).hom
[PROOFSTEP]
infer_instance
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : PreservesColimit (Functor.empty C) G
F : Discrete PEmpty ⥤ D
⊢ HasColimit F
[PROOFSTEP]
haveI := HasColimit.mk ⟨_, isColimitOfHasInitialOfPreservesColimit G⟩
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : PreservesColimit (Functor.empty C) G
F : Discrete PEmpty ⥤ D
this : HasColimit (Functor.empty D)
⊢ HasColimit F
[PROOFSTEP]
apply hasColimitOfIso F.uniqueFromEmpty
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : HasInitial D
i : IsIso (initialComparison G)
⊢ PreservesColimit (Functor.empty C) G
[PROOFSTEP]
apply preservesColimitOfPreservesColimitCocone initialIsInitial
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : HasInitial D
i : IsIso (initialComparison G)
⊢ IsColimit (G.mapCocone (asEmptyCocone (⊥_ C)))
[PROOFSTEP]
apply (isColimitMapCoconeEmptyCoconeEquiv _ _).symm _
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : HasInitial D
i : IsIso (initialComparison G)
⊢ IsInitial (G.obj (⊥_ C))
[PROOFSTEP]
exact @IsColimit.ofPointIso _ _ _ _ _ _ _ (colimit.isColimit (Functor.empty.{0} D)) i
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : HasInitial D
f : ⊥_ D ⟶ G.obj (⊥_ C)
i : IsIso f
⊢ PreservesColimit (Functor.empty C) G
[PROOFSTEP]
rw [Subsingleton.elim f (initialComparison G)] at i 
[GOAL]
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝¹ : HasInitial C
inst✝ : HasInitial D
f : ⊥_ D ⟶ G.obj (⊥_ C)
i : IsIso (initialComparison G)
⊢ PreservesColimit (Functor.empty C) G
[PROOFSTEP]
exact PreservesInitial.ofIsoComparison G
[GOAL]
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝² : HasInitial C
inst✝¹ : HasInitial D
inst✝ : PreservesColimit (Functor.empty C) G
⊢ IsIso (initialComparison G)
[PROOFSTEP]
rw [← PreservesInitial.iso_hom]
[GOAL]
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X : C
inst✝² : HasInitial C
inst✝¹ : HasInitial D
inst✝ : PreservesColimit (Functor.empty C) G
⊢ IsIso (PreservesInitial.iso G).inv
[PROOFSTEP]
infer_instance
