The definition about Galois connection at wiki is that
Let $(A, \leq)$ and $(B, \leq)$ be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions: $F\colon A \rightarrow B$ and $G\colon B \rightarrow A$, such that for all $a$ in $A$ and $b$ in $B$, we have $$F(a) \leq b~\text{if and only if}~ a \leq G(b).$$
The definition about Galois connection at nlab is that
Given posets $A$ and $B$, a Galois connection between $A$ and $B$ is a pair of order-reversing functions $f\colon{A}\rightarrow{B}$ and $g\colon{B}\rightarrow{A}$ such that $a \leq g(f(a))$ and $b \leq f(g(b))$ for all $a \in A$, $b \in B$.
I think they are equivalent, so i start proof it, but there is problem: suppose $(f,g)$ is a galois connection statisfies nlab definition, then i assume $a \leq g(b)$ and want to proof $f(a) \leq b$. we have $f(a) \leq f(g(b))$, but nlab's definition told us $b \leq f(g(b))$, so i can't go on.
i think the $b \leq f(g(b))$ should be changed to $f(g(b)) \leq b$. i dont know its right?
As explained in their very first paragraph, the definition on nLab refers to an antitone Galois connection, not monotone as on Wikipedia. So all inequalities on the “B” side are reversed.