Theorems · Definition · category theory

CategoryTheory.MorphismProperty.Comma.mapLeftComp

{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] →
            (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] →
                        {L₁ L₂ L₃ : CategoryTheory.Functor A T} →
                          [Q.RespectsIso] →
                            [W.RespectsIso] →
                              (l : L₁ ⟶ L₂) →
                                (l' : L₂ ⟶ L₃) →
                                  (hl :
                                      ∀ (X : CategoryTheory.MorphismProperty.Comma L₂ R P Q W),
                                        P (CategoryTheory.CategoryStruct.comp (l.app X.left) X.hom)) →
                                    (hl' :
                                        ∀ (X : CategoryTheory.MorphismProperty.Comma L₃ R P Q W),
                                          P (CategoryTheory.CategoryStruct.comp (l'.app X.left) X.hom)) →
                                      (hll' :
                                          ∀ (X : CategoryTheory.MorphismProperty.Comma L₃ R P Q W),
                                            P
                                              (CategoryTheory.CategoryStruct.comp
                                                ((CategoryTheory.CategoryStruct.comp l l').app X.left) X.hom)) →
                                        CategoryTheory.MorphismProperty.Comma.mapLeft R
                                            (CategoryTheory.CategoryStruct.comp l l') hll' ≅
                                          (CategoryTheory.MorphismProperty.Comma.mapLeft R l' hl').comp
                                            (CategoryTheory.MorphismProperty.Comma.mapLeft R l hl)

The functor P.Comma L₁ R Q W ⥤ P.Comma L₃ R Q W induced by the composition of two natural transformations l : L₁ ⟶ L₂ and l' : L₂ ⟶ L₃ is naturally isomorphic to the composition of the two 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