-- Mathlib/CategoryTheory/Limits/Presheaf.lean
theorem final_toCostructuredArrow_comp_pre {c : Cocone (F ⋙ yoneda)} (hc : IsColimit c) :
    Functor.Final (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt)

Proof state:
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
⊢ IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda))

Tactic:
suffices IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙
    CostructuredArrow.toOver yoneda c.pt)) by
  apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse
  apply IsTerminal.ofIso this
  refine ?_ ≪≫ (preservesColimitIso (overEquivPresheafCostructuredArrow c.pt).inverse _).symm
  apply HasColimit.isoOfNatIso
  exact isoWhiskerLeft _
    (CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow c.pt).isoCompInverse

Result:
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
⊢ IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
⊢ (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt).Final

Tactic:
apply Functor.final_of_isTerminal_colimit_comp_yoneda

Result:
case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
⊢ IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda))

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda))

Tactic:
apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse

Result:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ IsTerminal
    ((overEquivPresheafCostructuredArrow c.pt).inverse.obj
      (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda)))

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
isc : IsColimit
  ((Over.forget c.pt).mapCocone
    (colimit.cocone
      ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))) :=
  isColimitOfPreserves (Over.forget c.pt)
    (colimit.isColimit
      ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
i : I
⊢ c.ι.app i ≫
      (hc.coconePointUniqueUpToIso isc).hom ≫
        (colimit
            ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙
              CostructuredArrow.toOver yoneda c.pt)).hom =
    c.ι.app i ≫ (Over.mk (𝟙 c.pt)).hom

Tactic:
simp

Result:
no goals

Proof state:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt) ≅
    (overEquivPresheafCostructuredArrow c.pt).inverse.obj
      (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda))

Tactic:
refine ?_ ≪≫ (preservesColimitIso (overEquivPresheafCostructuredArrow c.pt).inverse _).symm

Result:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt) ≅
    colimit
      (((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda) ⋙
        (overEquivPresheafCostructuredArrow c.pt).inverse)

Proof state:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ IsTerminal
    ((overEquivPresheafCostructuredArrow c.pt).inverse.obj
      (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda)))

Tactic:
apply IsTerminal.ofIso this

Result:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt) ≅
    (overEquivPresheafCostructuredArrow c.pt).inverse.obj
      (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda))

Proof state:
case l.w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt ≅
    ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda) ⋙
      (overEquivPresheafCostructuredArrow c.pt).inverse

Tactic:
exact isoWhiskerLeft _
  (CostructuredArrow.toOverCompOverEquivPresheafCostructuredArrow c.pt).isoCompInverse

Result:
no goals

Proof state:
case l
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt) ≅
    colimit
      (((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda) ⋙
        (overEquivPresheafCostructuredArrow c.pt).inverse)

Tactic:
apply HasColimit.isoOfNatIso

Result:
case l.w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
I : Type v₁
inst✝ : SmallCategory I
F : I ⥤ C
c : Cocone (F ⋙ yoneda)
hc : IsColimit c
this :
  IsTerminal
    (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))
⊢ (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt ≅
    ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ yoneda) ⋙
      (overEquivPresheafCostructuredArrow c.pt).inverse