I once read (I believe in Ravi Vakil's notes on Algebraic Geometry) that the connected sum of a pair of surfaces can be defined in terms of a universal property. This gives a slick proof that the connected sum is unique up to homeomorphism. Unfortunately, I am unable to find where exactly I read this or remember what exactly universal property was; if anyone could help me out in either regard it would be much appreciated.
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As already noted in the comments, there is an obvious universal property (since the connected sum is a special pushout) once the embeddings of the discs have been chosen. For different embeddings, there exists some homeomorphism. There are lots of them, but even abstract nonsense cannot replace the nontrivial proof of existence. But since there is no canonical homeomorphism, I strongly doubt that there is a ny universal property which does not depend on the embeddings of the discs.