Is this always a Galois connection?

Question

In continuation of this question:

Suppose that we know that (for every elements $a$, $b$ of some posets) both:

• $F(a) = \inf \{ c \mid a \leq G(c) \}$;
• $G(b) = \sup \{ c \mid b \geq F(c) \}$.

Does it follow that $F$ and $G$ form a Galois connection between these posets?

Answer 1

Here is one direction

  f a ≤ b
≡    “set membership”
  a ∈ { c ∣ f c ≤ b }
⇒    “sup is an upper bound: e ∈ S ⇒ e ≤ sup S”
  a ≤ sup { c ∣ f c ≤ b }
≡    “definition of g” 
  a ≤ g b

now you do the other ;)