“function monotonicity” condition is required in definition of Galois connection?

Question

In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf definition of Galois connection is:
A Galois connection between preorders P and Q is a pair of monotone maps $ f:P→Q $ and $ g:Q→P $ such that: $$ f(p) ≤ q \iff p ≤ g(q) $$ We say that $f$ is the left adjoint and $g$ is the right adjoint of the Galois connection.

In this definition, monotonicity of $f$ and $g$ is condition. But I think with $ f(p) ≤ q \iff p ≤ g(q) $, it's suffice to prove monotonicity of $f$ and $g$.
So the monotonicity condition can be removed right?

Answer 1

That's right! A nice observation. From the condition we can conclude $p \le g(f(p))$, so if $p'\le p$ then $p'\le g(f(p)),$ and thus $f(p')\le f(p)$, so $f$ must be monotone. It's just rare enough to think about non-monotone maps of posets that this is easy to skip over.