English
LanUnit along Yoneda provides leftKanExtension in presheaf context with composition laws.
Русский
LanUnit вдоль Янеда обеспечивает leftKanExtension в контексте прешефа с законами композиции.
LaTeX
$$$\text{LanUnit}_{\yoneda}^{A}$ yields aLeftKanExtension$$
Lean4
@[reassoc (attr := simp)]
theorem coconeApp_naturality {P : Cᵒᵖ ⥤ Type max w v₁ v₂} {x y : P.Elements} (f : x ⟶ y) :
uliftYoneda.map f.1.unop ≫ coconeApp.{w} φ x = coconeApp φ y :=
by
have eq₁ : uliftYoneda.map f.1.unop ≫ uliftYonedaEquiv.symm x.2 = uliftYonedaEquiv.{max w v₂}.symm y.2 :=
uliftYonedaEquiv.injective (by simpa only [Equiv.apply_symm_apply, ← uliftYonedaEquiv_naturality] using f.2)
have eq₂ :=
congr_fun ((G.map (uliftYonedaEquiv.{max w v₂}.symm x.2)).naturality (F.map f.1.unop).op)
((φ.app x.1.unop).app _ (ULift.up (𝟙 _)))
have eq₃ := congr_fun (congr_app (φ.naturality f.1.unop) _) (ULift.up (𝟙 _))
have eq₄ := congr_fun ((φ.app x.1.unop).naturality (F.map f.1.unop).op)
dsimp at eq₂ eq₃ eq₄
apply uliftYonedaEquiv.{max w v₂}.injective
dsimp only [coconeApp]
rw [Equiv.apply_symm_apply, ← uliftYonedaEquiv_naturality, Equiv.apply_symm_apply]
simp only [← eq₁, ← eq₂, ← eq₃, ← eq₄, op_unop, Functor.comp_obj, Functor.op_obj, yoneda_obj_obj, Functor.comp_map,
Functor.op_map, Functor.map_comp, FunctorToTypes.comp, ]
simp [uliftYoneda]
