ag.algebraic geometry - About a non-obvious (?) link between the jacobians of curves and differentials

To explain my problem, I must give a lemma:

Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $pi : X to Y$, $phi : X to Z$ two non-constant morphisms.
If $phi^star(H^0(Z,Omega))subseteqpi^star(H^0(Y,Omega))$, where $Omega$ denotes the sheaf of regular 1-forms in each case, then there exists a non-constant morphism $u: Y to Z$ such that $phi = u circ pi$.

Now, in a proof, I saw the use of this lemma, except that the hypothesis was the inclusion $mathrm{Image}(mathrm{Jac}(Z) to mathrm{Jac}(X)) subseteq mathrm{Image}(mathrm{Jac}(Y) to mathrm{Jac}(X))$, instead of $phi^star(H^0(Z,Omega))subseteqpi^star(H^0(Y,Omega))$. I can guess it is equivalent, but why? Is it related to Grothendieck's duality? Did I miss something obvious?