I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \equiv 0, 1 \pmod 4$ to be the set $C(D)$ of classes of primitive positive definite integral binary quadratic forms (an assumption I make about all forms in this question) under proper equivalence, where two such forms $f, g$ are properly equivalent if there exists a matrix $\begin{pmatrix} p & q\\ r & s\\ \end{pmatrix} \in SL_2(\mathbb Z)$ for which $g(x, y) = f(px+qy, rx+sy)$. I know that the set of proper equivalence classes can be given a group structure under Dirichlet composition: where for two forms $f(x, y):= ax^2+bxy+cy^2$ and $g(x, y):= a'x^2+b'xy+c'y^2$, we define their Dirichlet composition to be the form $$F(x, y) := aa'x^2 + Bxy+ \frac{B^2-D}{4aa'}y^2$$ where $B$ is the unique integer in the interval $[0, 2aa')$ satisfying the following system of linear congruences $$B \equiv b \pmod{2a}, \hspace{5mm} B \equiv b' \pmod{2a'}, \hspace{5mm} B^2 \equiv D \pmod{4aa'}.$$
Most of the resources I could access give correspondence between the form class group and the whole ideal class group using the notion of orders (a notion that I am not comfortable with). However, I have been particularly interested in the concise treatment in these notes (https://www.renyi.hu/~gharcos/heegner.pdf) [Appendix, pages 384-386], which give a bijective correspondence between the form class group and the narrow class group (by the narrow class group, I mean the set of equivalence classes of fractional ideals under the equivalence relation $I \stackrel+\sim J$ if there exist totally positive $\alpha, \beta \in \mathcal O_K$ such that $\alpha I = \beta J$, with this set of classes given a group structure by multiplication), by considering a set of representatives $$Q_i(x, y) := a_i x^2 + b_i xy + c_i y^2 = a_i (x - z_i y) (x - \overline z_i y), \hspace{1mm} \text{ for all }i= 1, \cdots, h(D),$$ $$a_i>0, \hspace{1mm} z_i = \frac{-b_i+\sqrt D}{2 a_i}, \hspace{1mm} \overline z_i = \frac{-b_i-\sqrt D}{2 a_i},$$ of the equivalence classes of forms of fundamental discriminant $D$, and then demonstrating that each fractional ideal $I$ of $\mathbb Q(\sqrt D)$ is equivalent to one of the fractional ideals $$I_i = \mathbb Z + \mathbb Z z_i, \hspace{6mm} i= 1, \cdots, h(D),$$ while the distinct $I_i$ themselves are pairwise inequivalent.
The proof goes by considering an arbitrary fractional ideal $I = \mathbb Z \omega_1 + \mathbb Z \omega_2$ with $\overline \omega_1 \omega_2 - \omega_1 \overline \omega_2 > 0$ and associating to it, the binary quadratic form $$Q_I(x, y) := \frac{(x \omega_1 - y\omega_2)(x \overline \omega_1 - y \overline \omega_2)}{N(I)},$$ where $N(I)>0$ is the absolute norm of $I$. What I really like about the proof is that it is elementary and doesn't use notions of orders or congruence subgroups.
Now, I have been trying to see if I can motivate Dirichlet composition from this bijection by transport of structure (https://en.wikipedia.org/wiki/Transport_of_structure), but nothing I tried so far has worked. So my question is the following:
Can I show (without using any additional machinery such as orders or congruence subgroups, if possible) that for two ideals $I$ and $J$, the form $Q_{IJ}$ that gets associated to the product $IJ$ under the above correspondence is exactly the Dirichlet composition of $Q_I$ and $Q_J$?
I would be really grateful for suggestions, solutions or references where this has been worked out in detail (with the aforementioned restriction on the machinery used). Thank you.
