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.
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.
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