= Countable biproducts in an additive category force triviality
Suppose a <semi-additive category> has countable products and coproducts and the canonical comparison between them is always invertible. For countably many copies of $A$, let $q:A\to\coprod_iA$ correspond under that comparison to the diagonal into the product, and let $\sigma:\coprod_iA\to A$ be the fold map. If $s$ shifts the summands by one, coordinatewise comparison proves $q=\nu_1+sq$ and $\sigma s=\sigma$. Thus $z=\sigma q$ satisfies $z=1_A+z$. In an <additive category>, cancellation gives $1_A=0$, forcing every hom-set to contain only its zero morphism. The category is equivalent to the one-object, one-arrow category.
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