Let $\Gamma$ and $\Gamma'$ be congruence subgroups of $SL_2(\mathbb{Z})$, and $\alpha\in GL_2^+(\mathbb{Q})$. Then one defines a "double coset operator" or "change of automorphy operator" which sends $M_*(\Gamma)$ to $M_*(\Gamma')$ by sending a weight $k$ form $f$ to $\sum f|_k\beta_j$, where $\Gamma\backslash\Gamma\alpha\Gamma'=\{\Gamma\beta_1,\dotsc,\Gamma\beta_n\}$. This is a standard construction, and it can be used to construct Hecke operators, diamond operators, tautological "restriction of subgroup" operators, and "extension of subgroup" trace operators.
My question is as follows: what kind of functoriality does this operation have? Put more simply, is there a nice description for the composition of these operators? Say I have three congruence subgroups $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, and I choose two matrices $\alpha,\beta\in GL_2^+(\mathbb{Q})$. I can compose $|_k\Gamma_1\alpha\Gamma_2$ and $|_k\Gamma_2\beta\Gamma_3$ to get a linear map $M_k(\Gamma_1)\to M_k(\Gamma_3)$. How can I describe this map? Is it also a double coset operator?
For context, I'm interested in describing the modular-equivariant structure of $M_*$ as the congruence subgroup varies, and this description should include all (or at least a lot) of the change-of-automorphy operators. Something like a generalization of the Hecke algebra, but it contains operators going between different subgroups. (A "Hecke algebroid", if you will.) The ideal would be to describe it as the action of a category whose objects correspond to congruence subgroups, but I don't know what this category's composition would look like. (I would also be okay with describing this as an algebra over some kind of $SL_2(\mathbb{Z})$-equivariant operad, but that's obviously more complicated.)