A function $f : \omega \to \omega$ satisfies the domination property if $f$ is (eventually) dominated by the function $g : n \mapsto \sup \{ t : (\exists e \le n)(\exists \sigma \in 2^n)(\exists \tau \succeq \sigma)[\tau \in W_{e, t+1} \setminus W_{e, t}]\}$. The significance of the domination property includes that a function without the domination property computes a 1-generic set.
I would like to show
If all functions $M$ computes satisfy the domination property, $M$ is generalized low 2 (GL2).
The proof, I suppose, should involve reducing the $\Sigma^0_2(M)$-complete set $\mathrm{Tot}^M = \{e : \Phi_e^M\text{ is total}\}$ to $(M \oplus 0')'$. It is easy to see that the complement of $\mathrm{Tot}^M$ is c.e. relative to $M' \le_T (M \oplus 0')'$, so it only remains to show that $\mathrm{Tot}^M$ is c.e. relative to $(M \oplus 0')'$. For this, I should come up with an $M$-computable function $f$ to appeal to the assumption, but I have no idea what $f$ to pick. I am aware of the characterization of GL2 sets by Martin, which sounds a little bit similar to the situation above but not quite.
How can I show that?
Actually, that can be proved by using Martin's characterization of GL2 sets: a set $A$ is GL2 iff there is a function $ g \le A \oplus 0'$ that dominates all $A$-computable functions. The function $g$ above is $0'$-computable, and $M \oplus 0'$-computable a fortiori, so the result follows.