Why is the mean meridional mass stream function multiplied by g/2π?

I was reading this material on the mean atmospheric meridional circulation and the definition of the meridional mass streamfunction: http://www.geo.cornell.edu/geology/faculty/Cook/Hadley_book.pdf (Pag 3 and 4).

The continuity equation in spherical coordinates and using pressure as the vertical coordinate reads:

$$ frac{1}{a cosphi}frac{partial u}{partial lambda}+frac{1}{a cosphi}frac{partial (v cosphi)}{partial phi}+frac{partial omega}{partial p}=0$$

Where $a$ is the Earth radius, $phi$ is the latitude, $lambda$ is the longitude, $u,v$ the horizontal wind components on a pressure level, $omega$ is the vertical pseudovelocity in pressure coordinates.
My first doubt is in fact when they state that the zonal (longitudinal) mean of this equation is

$$frac{1}{a}frac{partial[v]}{partial phi}+frac{partial[omega]}{partial p}=0$$

where [ ] indicates the zonal mean. I get why the first term $frac{1}{a cosphi}frac{partial u}{partial lambda} $ vanishes; I don’t get though why $frac{1}{a cosphi}frac{partial (v cosphi)}{partial phi}$ reduces to $frac{1}{a}frac{partial[v]}{partial phi} $. By the product differentiation rule, I’d expect it to look like

$$frac{1}{a}frac{partial[v]}{partial phi}-frac{tanphi}{a}[v]$$

especially because they then define the meridional streamfunction $Psi$ such that

$$ [v] = frac{g}{2pi a cosphi}frac{partialPsi}{partial p} $$

and

$$ [omega] = -frac{g}{2pi a^{2} cosphi}frac{partialPsi}{partial phi} $$

so that when we do $frac{1}{a}frac{partial[v]}{partial phi}+frac{partial[omega]}{partial p}$ using this definition it actually results in $frac{tanphi}{a}[v]$, not $0$.

I wonder if there is a mistake there or is there something I’m not following?

My second doubt is, why multiply by the factor $frac{g}{2pi}$ the definition of the streamfunction, since it will be cancelled out anyway when replacing this definition into the continuity equation? I mean, it doesn’t actually interfere in this definition satisfying the continuity equation or not.