This question may be kind of vague. And we use the **same** notations as in Carayol's papers:

*H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert;*

*H. Carayol, Sur la mauvaise réduction des courbes de Shimura.*

We know Carayol constructed l-adic representation $\sigma$ of $Gal(\bar{F}/F)$ "in" the étale cohomology group of "quaternionic" Shimura curves, roughly speaking, by taking some Hecke "eigenspace" of $H_{ét}^1(M_K\times_F \bar{F}, \mathcal{F}_{\lambda})$.

My question is could we get this Galois representation from some unitary Shimura curves? More precisely, does $\sigma|_{Gal(\bar{E}/E)}$ appear in $H_{ét}^1(M_{K'}'\times_E \bar{E},\mathcal{F}_{\lambda}')$ for some $K'$, and some locally constant sheaf $\mathcal{F}_{\lambda}'$?

Thanks!