This work is in the continuity of the first two authors’ work to build a theory of game models. It incorporates into this line of work a recent recasting of game semantics by Melliès in order to understand composition of strategies as an abstract categorical construction. Melliès’s framework is strikingly simple, and can be described as proceeding in two steps: - first, he designs a particular categorical structure, called the “template”, to describe the scheduling policy of strategies in a certain type of games (technically, a monad internal to a certain weak double category Span(Cat)), and the striking point is that its definition precisely follows basic game semantical intuitions, - then, the bicategory of games, strategies, and simulations is simply obtained as a sort of slicing construction over the template in Span(Cat) (this construction is just a double-categorical variant of the standard slice construction). However, this simple construction does not yield exactly standard games.

We analyse what makes Melliès’s games non-standard, and recover simple games by first slightly modifying the template, and then generalising the slice construction by introducing the use of a factorisation system. This construction can be done at a high level of generality (technically, under suitable conditions, this slicing can be done using any monad M internal to a weak double category D equipped with a factorisation system (L,R) to derive a new weak double category by slicing), and we show how to apply it to recover both simple game semantics and Day convolution as instances. This in passing explains the similarity between composition of strategies and Day convolution.