Theorems · Definition · category theory

CategoryTheory.LaxMonoidalFunctor.isoOfComponents

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    [inst_1 : CategoryTheory.MonoidalCategory C] →
      {D : Type u₂} →
        [inst_2 : CategoryTheory.Category.{v₂, u₂} D] →
          [inst_3 : CategoryTheory.MonoidalCategory D] →
            {F G : CategoryTheory.LaxMonoidalFunctor C D} →
              (e : (X : C) → F.obj X ≅ G.obj X) →
                autoParam
                    (∀ {X Y : C} (f : X ⟶ Y),
                      CategoryTheory.CategoryStruct.comp (F.map f) (e Y).hom =
                        CategoryTheory.CategoryStruct.comp (e X).hom (G.map f))
                    CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_1 →
                  autoParam
                      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.ε F.toFunctor)
                          (e (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).hom =
                        CategoryTheory.Functor.LaxMonoidal.ε G.toFunctor)
                      CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_3 →
                    autoParam
                        (∀ (X Y : C),
                          CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.μ F.toFunctor X Y)
                              (e (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)).hom =
                            CategoryTheory.CategoryStruct.comp
                              (CategoryTheory.MonoidalCategoryStruct.tensorHom (e X).hom (e Y).hom)
                              (CategoryTheory.Functor.LaxMonoidal.μ G.toFunctor X Y))
                        CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_5 →
                      (F ≅ G)

Constructor for isomorphisms between lax monoidal functors.

Defined in

Mathlib.CategoryTheory.Monoidal.NaturalTransformation

Cited by

4 results in Mathlib

Foundations

Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategory