Very good example + presentation! Maybe this example can be made to work as a counterexample for the claim "The composition of 2 universal maps is universal"

Well, the function space $X^Y$ can be defined for any topological spaces, inheriting the topology from the product topology..

Why do you think that it makes a difference if you restrict yourself to compactly generated spaces?

Hang on - I have a problem with your definition. If we take it to the category of posets, isn't every order-preserving universal (in your categorical definition) if we can take $Z = \emptyset$? (There is exactly one order preserving map from $\emptyset$ to any poset).

Very nice! If you have time, you can copy the definition above as an answer to mathoverflow.net/questions/189377/…

I was wondering, is there a universal property in "category theory language" that captures the essence of a universal map? - I guess, I better make this a real question (with a category-theory tag).