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Mathlib.CategoryTheory.Monoidal.Cartesian.Comon_

Comonoid objects in a cartesian monoidal category. #

The category of comonoid objects in a cartesian monoidal category is equivalent to the category itself, via the forgetful functor.

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def cartesianComon_ (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
CategoryTheory.Functor C (Comon_ C)

The functor from a cartesian monoidal category to comonoids in that category, equipping every object with the diagonal map as a comultiplication.

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    @[simp]
    theorem counit_eq_from {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] (A : Comon_ C) :
    A.counit = CategoryTheory.Limits.terminal.from A.X
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    @[simp]
    theorem comul_eq_diag {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] (A : Comon_ C) :
    A.comul = CategoryTheory.Limits.diag A.X
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    @[simp]
    theorem iso_cartesianComon__hom_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] (A : Comon_ C) :
    (iso_cartesianComon_ A).hom.hom = CategoryTheory.CategoryStruct.id A.X
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    @[simp]
    theorem iso_cartesianComon__inv_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] (A : Comon_ C) :
    (iso_cartesianComon_ A).inv.hom = CategoryTheory.CategoryStruct.id ((cartesianComon_ C).obj A.X).X
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    def iso_cartesianComon_ {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] (A : Comon_ C) :
    A ≅ (cartesianComon_ C).obj A.X

    Every comonoid object in a cartesian monoidal category is equivalent to the canonical comonoid structure on the underlying object.

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      @[simp]
      theorem comonEquiv_unitIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
      comonEquiv.unitIso = CategoryTheory.NatIso.ofComponents (fun (A : Comon_ C) => iso_cartesianComon_ A) ⋯
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      @[simp]
      theorem comonEquiv_functor {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
      comonEquiv.functor = Comon_.forget C
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      @[simp]
      theorem comonEquiv_inverse {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
      comonEquiv.inverse = cartesianComon_ C
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      @[simp]
      theorem comonEquiv_counitIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
      comonEquiv.counitIso = CategoryTheory.NatIso.ofComponents (fun (X : C) => CategoryTheory.Iso.refl (((cartesianComon_ C).comp (Comon_.forget C)).obj X)) ⋯
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      def comonEquiv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasBinaryProducts C] :
      Comon_ C ≌ C

      The category of comonoid objects in a cartesian monoidal category is equivalent to the category itself, via the forgetful functor.

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