preservesEqualizers_of_preservesKernels

English

For every natural n, I.ι.f n is mono (reiterating the mono property in Nat-indexed family).

Русский

Для каждого натурального n множество I.ι.f n моно; повторение моно свойства для семейства с индексами по Nat.

LaTeX

$$$\forall n\in\mathbb{N},\; Mono(I.ι.f\,n)$$$

Lean4

/-- A functor between preadditive categories preserves all equalizers if it preserves all kernels.
-/
theorem preservesEqualizers_of_preservesKernels [∀ {X Y} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F] :
    PreservesLimitsOfShape WalkingParallelPair F where
  preservesLimit
    {K} :=
    by
    letI :=
      preservesEqualizer_of_preservesKernels F (K.map WalkingParallelPairHom.left) (K.map WalkingParallelPairHom.right)
    apply preservesLimit_of_iso_diagram F (diagramIsoParallelPair K).symm

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