Theorems · Definition · category theory

CategoryTheory.MorphismProperty.Comma.mapLeftId

{A : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} A] →
    {B : Type u_2} →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} B] →
        {T : Type u_3} →
          [inst_2 : CategoryTheory.Category.{v_3, u_3} T] →
            (L : CategoryTheory.Functor A T) →
              (R : CategoryTheory.Functor B T) →
                {P : CategoryTheory.MorphismProperty T} →
                  {Q : CategoryTheory.MorphismProperty A} →
                    {W : CategoryTheory.MorphismProperty B} →
                      [inst_3 : Q.IsMultiplicative] →
                        [inst_4 : W.IsMultiplicative] →
                          [Q.RespectsIso] →
                            [W.RespectsIso] →
                              CategoryTheory.MorphismProperty.Comma.mapLeft R (CategoryTheory.CategoryStruct.id L) ⋯ ≅
                                CategoryTheory.Functor.id (CategoryTheory.MorphismProperty.Comma L R P Q W)

The functor P.Comma L R Q W ⥤ P.Comma L R Q W induced by the identity natural transformation on L is naturally isomorphic to the identity functor.

Defined in

Mathlib.CategoryTheory.MorphismProperty.Comma

Cited by

5 results in Mathlib

Foundations

Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.MorphismProperty.RespectsIsoCategoryTheory.MorphismProperty.RespectsIso