(This is very vague, so sorry if there are approximations)
I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a filter of x to a filter of f(x)
Do you know similar examples in other areas, like probability, where such seemingly analytical properties can likewise be expressed as commutation on structures but are usually not so ?
I vaguely believe there was some in measure theory.