  We use the adelic language to show that any homomorphism between Jacobians of
modular curves arises from a linear combination of Hecke modular
correspondences. The proof is based on a study of the actions of
$\mathrm{GL}_2$ and Galois on the \'etale cohomology of the tower of modular
curves. We also make this result explicit for Ribet's twisting operators on
modular abelian varieties.
