Induced map on jacobians of curves of genus one

Question

Let $K$ be a field and consider two genus-one curves $C,C^{\prime}$ (probably without $K$-points). Let $f:C\to C^{\prime} $ be a morphism defined over $K$. Do I understand correctly that if we take the induced map on jacobians $J(C^{\prime})\to J ( C)$ and than consider its dual $J ( C)\to J(C^{\prime})$ we obtain an isogeny of elliptic curves over $K$ isomorphic to $f$ over $\overline{K}$?