The trigonal construction in the ramified case

To every double cover ramified in two points of a general trigonal curve of genus g$g$, one can associate an étale double cover of a tetragonal curve of genus g+1$g+1$. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties. This extends Recillas' trigonal construction to covers ramified in two points.