preservesBinaryProduct_of_preservesKernels

English

A kernel-preserving functor between preadditive categories preserves the binary product of any pair of objects.

Русский

Кернел-предохранющаяфункция между прeддвативными категориями сохраняет двоичное произведение пары объектов.

LaTeX

$$$\forall X,Y\,\ Pr oduct(X,Y)\implies PreservesLimit(\text{pair }X\ Y, F)$$$

Lean4

/-- A kernel-preserving functor between preadditive categories preserves any pair being a limit. -/
theorem preservesBinaryProduct_of_preservesKernels [∀ {X Y} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F]
    {X Y : C} : PreservesLimit (pair X Y) F where
  preserves {c}
    hc :=
    ⟨IsLimit.ofIsoLimit (isLimitMapConeBinaryFanOfPreservesKernels F _ _ (IsLimit.ofIsoLimit hc (isoBinaryFanMk c)))
        ((Cones.functoriality _ F).mapIso (isoBinaryFanMk c).symm)⟩

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