Theorems · Definition · category theory

CategoryTheory.Limits.limitObjIsoLimitCompEvaluation

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {J : Type u₁} →
      [inst_1 : CategoryTheory.Category.{v₁, u₁} J] →
        {K : Type u₂} →
          [inst_2 : CategoryTheory.Category.{v₂, u₂} K] →
            [inst_3 : CategoryTheory.Limits.HasLimitsOfShape J C] →
              (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) →
                (k : K) →
                  (CategoryTheory.Limits.limit F).obj k ≅
                    CategoryTheory.Limits.limit (F.comp ((CategoryTheory.evaluation K C).obj k))

If F : J ⥤ K ⥤ C is a functor into a functor category which has a limit, then the evaluation of that limit at k is the limit of the evaluations of F.obj j at k.

Defined in

Mathlib.CategoryTheory.Limits.FunctorCategory.Basic

Cited by

36 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasLimitsOfShape