Theorems · Definition · category theory

CategoryTheory.MorphismProperty.Comma.mapRightComp

{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) →
              {P : CategoryTheory.MorphismProperty T} →
                {Q : CategoryTheory.MorphismProperty A} →
                  {W : CategoryTheory.MorphismProperty B} →
                    [inst_3 : Q.IsMultiplicative] →
                      [inst_4 : W.IsMultiplicative] →
                        {R₁ R₂ R₃ : CategoryTheory.Functor B T} →
                          [Q.RespectsIso] →
                            [W.RespectsIso] →
                              (r : R₁ ⟶ R₂) →
                                (r' : R₂ ⟶ R₃) →
                                  (hr :
                                      ∀ (X : CategoryTheory.MorphismProperty.Comma L R₁ P Q W),
                                        P (CategoryTheory.CategoryStruct.comp X.hom (r.app X.right))) →
                                    (hr' :
                                        ∀ (X : CategoryTheory.MorphismProperty.Comma L R₂ P Q W),
                                          P (CategoryTheory.CategoryStruct.comp X.hom (r'.app X.right))) →
                                      (hrr' :
                                          ∀ (X : CategoryTheory.MorphismProperty.Comma L R₁ P Q W),
                                            P
                                              (CategoryTheory.CategoryStruct.comp X.hom
                                                ((CategoryTheory.CategoryStruct.comp r r').app X.right))) →
                                        CategoryTheory.MorphismProperty.Comma.mapRight L
                                            (CategoryTheory.CategoryStruct.comp r r') hrr' ≅
                                          (CategoryTheory.MorphismProperty.Comma.mapRight L r hr).comp
                                            (CategoryTheory.MorphismProperty.Comma.mapRight L r' hr')

The functor P.Comma L R₁ Q W ⥤ P.Comma L R₃ Q W induced by the composition of the natural transformations r : R₁ ⟶ R₂ and r' : R₂ ⟶ R₃ is naturally isomorphic to the composition of the functors induced by these natural transformations.

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