Why is the relation between luminosity distance $d_L$ and comoving distance $chi$ $d_L=chi/a$?

The textbook (Scott Dodelson, Modern Cosmology, Section 2.2 Distance, Page 35-36) states as follows:

Another way of inferring distances in astronomy is to measure the flux from
an object of known luminosity. Recall that (forgetting about expansion for the
moment) the observed flux $F$ a distance d from a source of known luminosity $L$ is
begin{equation}
F=frac{L}{4pi d^2} tag{2.47}
end{equation}

since the total luminosity through a spherical shell with area $4pi d^2$ is
constant. How does this result generalize to an expanding universe? Again it is simplest to work on the comoving grid, this time with the source centered at the origin. The flux we observe is
begin{equation}
F=frac{L(chi)}{4pi chi^2 (a)} tag{2.48}
end{equation}
where $L(chi)$ is the luminosity through a (comoving) spherical shell with radius $chi(a)$. To further simplify, let’s assume that the photons are all emitted with the same energy. Then $L(chi)$ is this energy multiplied by the number of photons passing through a (comoving) spherical shell per unit time. In a fixed time interval, photons travel farther on the comoving grid at early times than at late times since the associated physical distance at early times is smaller. Therefore, the number of photons crossing a shell in the fixed time interval will be smaller today than at emission, smaller by a factor of $a$. Similarly, the energy of the photons will be smaller today than at emission, because of expansion. Therefore, the energy per unit time passing through a comoving shell a distance $chi(a)$ (i.e., our distance) from the source will be a factor of $a^2$ smaller than the luminosity at the source. The flux we observe therefore will be
begin{equation}
F=frac{La^2}{4pi chi^2 (a)} tag{2.49}
end{equation}
where $L$ is the luminosity at the source. We can keep Eq. (2.47) in an expanding
universe as long as we define the luminosity distance
begin{equation}
d_Lequivchi/a tag{2.50}
end{equation}

The questions that bothers me are

1. According to the Dodelson’s statements, there should be $L(chi)propto frac{1}{a^2}$. Why does $L(chi)$ equal to $La^2$ in Eq 2.49 ?
2. To my understanding, the physical distance is the comoving distance multiplied scale factor, i.e. $d=acdot chi$. But, Eq 2.50 violates it obviously. Why?