A double category of games

Question

Can recursive-coalgebra maps and playing strategies be combined in a double category of games? The intended vertical maps are exact coalgebra morphisms; the horizontal maps should come from an explicitly chosen category of games and strategies in the tradition of Joyal.

Current boundary

A coalgebra map preserves the complete set of options. A playing strategy chooses responses along legal plays. These are different structures, and their interaction must be defined rather than identified.

The work in note 13 shows why a counterexample with arbitrary binary relations cannot automatically be treated as a counterexample with legal total deterministic strategies. The elementary relational obstruction is recorded in note 23; it does not settle this double-category question.

Next step

Fix the objects (including any Left/Right or turn data), the legal strategy class, and the proposed squares. On a small example, verify that vertical maps transport legal strategies and that horizontal composition is compatible with the squares. State any restrictions needed on the vertical maps.

A construction must verify units, both compositions, and interchange. A no-go claim must exhibit a counterexample inside the chosen legal strategy class.

Sources: Games as recursive coalgebras; Joyal's games and strategies.