Keith: there's nothing wrong with your reasoning! So, now I understand what you mean and all is well.

However, it's not good to say "a monotone map \\(f : X \to Y \\)" here. For one thing, when you start a conversation by saying "a monotone map \\(f : X \to Y \\)," you are leaving it to us to guess what \\(X\\) and \\(Y\\) are. The two immediate guesses are "arbitrary preorders" or "arbitrary posets", since that's what monotone maps go between... but of course, other things in what you wrote pull us toward guessing that \\(X\\) and \\(Y\\) are sets, so we're left scratching our heads. Only a few very sneaky mathematicians would guess that you meant \\(X\\) and \\(Y\\) are sets being treated as discrete posets.

It's easier to understand "a function \\(f : X \to Y\\)", and even easier to understand "a function \\(f\\) from the set \\(X\\) to the set \\(Y\\)".

However, it's not good to say "a monotone map \\(f : X \to Y \\)" here. For one thing, when you start a conversation by saying "a monotone map \\(f : X \to Y \\)," you are leaving it to us to guess what \\(X\\) and \\(Y\\) are. The two immediate guesses are "arbitrary preorders" or "arbitrary posets", since that's what monotone maps go between... but of course, other things in what you wrote pull us toward guessing that \\(X\\) and \\(Y\\) are sets, so we're left scratching our heads. Only a few very sneaky mathematicians would guess that you meant \\(X\\) and \\(Y\\) are sets being treated as discrete posets.

It's easier to understand "a function \\(f : X \to Y\\)", and even easier to understand "a function \\(f\\) from the set \\(X\\) to the set \\(Y\\)".