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Certain nonlocal games exhibiting quantum advantage, such as the quantum graph homomorphism and isomorphism games, have composable quantum strategies which are naturally interpreted as structure-preserving functions between finite sets. We propose a natural compositional framework for noncommutative finite set theory in which these quantum strategies appear naturally, and which connects nonlocal games with recent work on compact quantum groups. We apply Morita-theoretical machinery within this framework to characterise, classify, and construct quantum strategies for the graph isomorphism game. This is joint work with Benjamin Musto and David Reutter, based on the papers 1711.07945 and 1801.09705.