Function Spaces in the Category of Directed Suprema Preserving Maps

Let F be a function. We say that F is uncurrying if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) For every set x such that x ∈ domF holds x is a function yielding function, and (ii) for every function f such that f ∈ domF holds F (f) = uncurry f. We say that F is currying if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) For every set x such that x ∈ domF holds x is a function and π1(x) is a binary relation, and (ii) for every function f such that f ∈ domF holds F (f) = curry f. We say that F is commuting if and only if the conditions (Def. 3) are satisfied.