Theorems · Theorem · category theory

CategoryTheory.Limits.equalizerPullbackMapIso_hom_snd

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasEqualizers C]
  [inst_2 : CategoryTheory.Limits.HasPullbacks C] {X Y S T : C} {f g : X ⟶ Y} {s : X ⟶ S} {t : Y ⟶ S}
  (hf : CategoryTheory.CategoryStruct.comp f t = s) (hg : CategoryTheory.CategoryStruct.comp g t = s) (v : T ⟶ S),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.equalizerPullbackMapIso hf hg v).hom
      (CategoryTheory.Limits.pullback.snd (CategoryTheory.Limits.equalizer.ι f g)
        (CategoryTheory.Limits.pullback.fst s v)) =
    CategoryTheory.Limits.equalizer.ι
      (CategoryTheory.Limits.pullback.map s v t v f (CategoryTheory.CategoryStruct.id T)
        (CategoryTheory.CategoryStruct.id S) ⋯ ⋯)
      (CategoryTheory.Limits.pullback.map s v t v g (CategoryTheory.CategoryStruct.id T)
        (CategoryTheory.CategoryStruct.id S) ⋯ ⋯)

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer

Cited by

1 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasEqualizersCategoryTheory.Limits.HasPullbacks