Theorems · Theorem · category theory

CategoryTheory.Adjunction.CoreUnitCounit.mk.inj

∀ {C : Type u₁} {inst : CategoryTheory.Category.{v₁, u₁} C} {D : Type u₂} {inst_1 : CategoryTheory.Category.{v₂, u₂} D}
  {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {unit : CategoryTheory.Functor.id C ⟶ F.comp G}
  {counit : G.comp F ⟶ CategoryTheory.Functor.id D}
  {left_triangle :
    autoParam
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight unit F)
          (CategoryTheory.CategoryStruct.comp (F.associator G F).hom (F.whiskerLeft counit)) =
        CategoryTheory.NatTrans.id ((CategoryTheory.Functor.id C).comp F))
      CategoryTheory.Adjunction.CoreUnitCounit.left_triangle._autoParam}
  {right_triangle :
    autoParam
      (CategoryTheory.CategoryStruct.comp (G.whiskerLeft unit)
          (CategoryTheory.CategoryStruct.comp (G.associator F G).inv (CategoryTheory.Functor.whiskerRight counit G)) =
        CategoryTheory.NatTrans.id (G.comp (CategoryTheory.Functor.id C)))
      CategoryTheory.Adjunction.CoreUnitCounit.right_triangle._autoParam}
  {unit_1 : CategoryTheory.Functor.id C ⟶ F.comp G} {counit_1 : G.comp F ⟶ CategoryTheory.Functor.id D}
  {left_triangle_1 :
    autoParam
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight unit_1 F)
          (CategoryTheory.CategoryStruct.comp (F.associator G F).hom (F.whiskerLeft counit_1)) =
        CategoryTheory.NatTrans.id ((CategoryTheory.Functor.id C).comp F))
      CategoryTheory.Adjunction.CoreUnitCounit.left_triangle._autoParam}
  {right_triangle_1 :
    autoParam
      (CategoryTheory.CategoryStruct.comp (G.whiskerLeft unit_1)
          (CategoryTheory.CategoryStruct.comp (G.associator F G).inv (CategoryTheory.Functor.whiskerRight counit_1 G)) =
        CategoryTheory.NatTrans.id (G.comp (CategoryTheory.Functor.id C)))
      CategoryTheory.Adjunction.CoreUnitCounit.right_triangle._autoParam},
  { unit := unit, counit := counit, left_triangle := left_triangle, right_triangle := right_triangle } =
      { unit := unit_1, counit := counit_1, left_triangle := left_triangle_1, right_triangle := right_triangle_1 } →
    unit = unit_1 ∧ counit = counit_1

Defined in

Mathlib.CategoryTheory.Adjunction.Basic

Cited by

1 results in Mathlib

Foundations

Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound