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What is the best characterization of categories which have all the products? Is this question related to Axiom of choice in any way?

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    There is no relation to the axiom of choice. A category with _all_ products is necessarily a preorder by a result of Freyd: see [this question](http://math.stackexchange.com/questions/202852/categories-with-limits-for-large-diagrams).2012-10-23

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