  Game semantics has provided adequate models for a variety of programming
languages, in which types are interpreted as two-player games and programs as
strategies. Melli\`es (2018) suggested that such categories of games and
strategies may be obtained as instances of a simple abstract construction on
weak double categories. However, in the particular case of simple games, his
construction slightly differs from the standard category. We refine the
abstract construction using factorisation systems, and show that the new
construction yields the standard category of simple games and strategies.
Another perhaps surprising instance is Day's convolution monoidal structure on
the category of presheaves over a strict monoidal category.
