I'm getting in a huge tangle with an innocuous-looking question about commuting of products and pushouts. I'm no good at this classical stuff.
You could probably modify this paper's proof that cartesian closed categories are distributive to handle pushouts. https://arxiv.org/abs/1406.0961
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Somehow I doubt it would be that simple, but this looks like a nice paper anyway, thanks!
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Doesn't the one-line proof of distributivity also imply preserving pushouts? Pushouts are colimits, after all.
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Nope! This confused me for quite a while. It gives you X x (B +_A C) isomorphic to (X x B) +_{X x A} (X x C). But I think I have a proof that goes via this.
End of conversation
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