luminosity - Conversion of magnitudes to Jansky and MAGPHYS?

I'm a bit puzzled and not an observer, so please bear with me if I'm being stupid here.

The code MAGPHYS specifies Jansky (Jy) as the input unit for flux through a filter (see Section 3.2.3 in the documentation). Meaning MAGPHYS wants me to use Jy as unit, when specify the total light of a whole galaxy in e.g. the SDSS-g band. But Jy is a unit of spectral density, i.e. W/m²/Hz. I would expect a flux as input unit when talking about the amont of light coming through a filter, i.e. in W/m² or in $L_odot$ (solar luminosity), so integrated over all frequencies that go through the filter and weighted by the filter response function.

If I have the absolute AB magnitude of a galaxy in the SDSS-g band, how do I convert it to Jy so that MAGPHYS will be happy?

I'm aware of:

$m_{text{AB}nu} = -2.5 log_{10} left(frac{f_{nu}}{3631text{Jy}}right) qquad text{or} qquad left(frac{f_{nu}}{text{Jy}}right) = 10^{-0.4 (m_{text{AB}nu} - 8.9)}$

But that is in one frequency, not a whole band.

Additionally, I have found the formula below in TOPCAT:

$left(frac{F}{text{Jy}}right) = 10^{left(23-0.4 (m_text{AB}+48.6)right)}$

Where does the 23 in the exponent come from?