Category of strategic games and presheaf corresponding to solution concepts or welfare criteria

We define a category of strategic games in which a morphism is a pair of a map between sets of players and a map between sets of strategy profiles with specific properties and show that this category is well-defined. Three cases are considered for the maps between sets of strategy profiles: they may be order-preserving, order-reflecting or order-embedding with respect to each player's preference relation. We define a presheaf on the category of games valued in a category of sets that sends each strategic game to a set of strategy profiles and present conditions for this presheaf to be well-defined. Two cases are considered for the morphisms in the category of sets: they may be relations or maps. We define a presheaf that sends each strategic game to the set of Nash equilibria (resp. Pareto efficient strategy profiles) and show that this presheaf is well-defined if and only if the maps between sets of strategy profiles are order-reflecting or order-embedding (resp. order-embedding), and the morphisms in the category of sets are relations.