I just realised that a Galois connection between posets that preserves meets and joins is already a bijection. This is easily proven as follows: If F: A -> B and G: B -> A is a Galois connection, then G(b) = min_{F(a) \ge b} a. Since F preserves the join, what G(b) is mapped to is >= b, but by the analogous property for F, it is also <=b. Since B is a poset, F(G(b)) = b, and the other direction is proven symmetrically.