Inspired by this question, does anyone know the strongest condition that ensures a category has sums if and only if it has products. What of other (co)limits?

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    $\begingroup$ "Weakest" implies the set of such conditions should be directed under implication, which is probably false. One nice condition that implies products and coproducts appear simultaneously is preadditivity. $\endgroup$ – Kevin Carlson Sep 24 '16 at 1:02
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    $\begingroup$ Sure, you stated it in your question: "it has products iff it has coproducts". That's literally by definition the "weakest" condition. $\endgroup$ – Najib Idrissi Sep 24 '16 at 9:40

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