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Dec 28, 2015 at 18:11 comment added André Henriques You are right. There is no way that I can see of recovering the notion "pair of transverse maps" given the notion "(surjective) submersion". So my answer is not a very good answer... Sorry for that.
Dec 28, 2015 at 4:26 comment added Mike Shulman This mentions the finite limits, split idempotents, and pullbacks of surjective submersions, but nothing else that I can see, e.g. not pullbacks of non-surjective submersions, or pullbacks of pairs of transverse maps neither of which is a submersion.
Dec 27, 2015 at 21:36 history answered André Henriques CC BY-SA 3.0