Proposition 1.81 (the basic theory of Galois connections) starts by supposing

that \\(f: P → Q\\) and \\(g: Q → P\\) are functions and shows that two things

are equivalent: a.) is that \\(f\\) and \\(g\\) form a Galois connection and b.)

is Eq. (1.6). Going from a.) to b.) is straightforward. In the proof of the

reverse, from b.) to a.), the first step in the proof goes: "then since \\(g\\)

is monotonic...". Is this a valid assumption? This is the first step is this

side of the proof and the basic setup only invites us to suppose that \\(f: P →

Q\\) and \\(g: Q → P\\) are functions.

that \\(f: P → Q\\) and \\(g: Q → P\\) are functions and shows that two things

are equivalent: a.) is that \\(f\\) and \\(g\\) form a Galois connection and b.)

is Eq. (1.6). Going from a.) to b.) is straightforward. In the proof of the

reverse, from b.) to a.), the first step in the proof goes: "then since \\(g\\)

is monotonic...". Is this a valid assumption? This is the first step is this

side of the proof and the basic setup only invites us to suppose that \\(f: P →

Q\\) and \\(g: Q → P\\) are functions.