of Eulerian solution, we are able to show that $\mu^F_\Psi$ and $\mu^L_\Psi$ are indeed the unique solutions to the system $$\begin{aligned}
\label{I8}
\left\{\begin{aligned}
\partial_t \mu^F_t& = -\mathrm{div}\big((K^{F}\star\mu^F_t + K^{L}\star\mu^L_t)\mu^F_t\big) - \alpha_F(x,\mu^F_t,\mu^L_t)\mu^F_t + \alpha_L(x, \mu^F_t,\mu^L_t)\mu^L_t,\\
\partial_t \mu^L_t& = -\mathrm{div}\big((K^{F}\star\mu^F_t + K^{L}\star\mu^L_t)\mu^L_t\big) +\alpha_F(x, \mu^F_t,\mu^L_t)\mu^F_t -\alpha_L(x, \mu^F_t,\mu^L_t)\mu^L_t
\end{aligned}\right.\end{aligned}$$ considered in [@ABRS2018]. Notice that, as discussed in Remark \[Tfin\], rewriting as a system of equation in $\mu^F$ and $\mu^L$ is not possible for a velocity field explicitly depending on $\lambda$ such as $$\begin{split}
v_{\Psi^N}(x_i, \lambda_i)=& \lambda_i(\{F\})\left(\frac{1}{N}\sum_{j=1}^N K^{FF}(x_i-x_j)\lambda_j(\{F\})+\frac{1}{N}\sum_{j=1}^N K^{LF}(x_i-x_j)\lambda_j(\{L\})\right)+\\
&\lambda_i(\{L\})\left(\frac{1}{N}\sum_{j=1}^N K^{FL}(x_i-x_j)\lambda_j(\{F\})+\frac{1}{N}\sum_{j=1}^N K^{LL}(x_i-x_j)\lambda_j(\{L\})\right)
\end{split}$$ considered for instance in [@francesco2013measure]. In the presence of exchange rates, the correct mean-field description of the above particle system is given by in the product space $\mathbb{R}^d\times {\mathcal P}_1(\{F, L\})$.

The paper is organized as follows. In Section \[sec:prel\], we recall the basic notions of measure theory that will be needed in the sequel, and prove a corollary of a theorem by Brezis [@Brezis Sect. I.3, Thm. 1.4, Cor. 1.1] on the well-posedness of ODE’s in Banach spaces. In Section \[sec:abstract\_mod\] we introduce our abstract model and we prove our main result, Theorem \[T150\], on the mean-field limit of the dynamics. Section \[sec:LF\] is devoted to the special case of $U=\{F,L\}$, 