= Coproduct-to-product comparison in a finitary abelian category
{title2=$j:\coprod_iA_i\to\prod_iA_i$}
In a complete <finitary abelian category>, the comparison $j$ with diagonal matrix entries the identity and off-diagonal entries zero is monic. Express its source as the directed colimit of finite <biproducts>. Their maps into the product have finite-coordinate retractions and are therefore <split monomorphisms>; the finitary axiom applies to this cocone. If the category is also <cofinitary abelian category>, duality makes $j$ epic and hence invertible. For countably many copies of any object, the shift-and-fold argument in <countable biproducts in an additive category force triviality> then forces the identity to be zero, so every object is a zero object.
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