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 $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$, 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 $\phi = u \circ \pi$.
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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 $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$. I can guess it is equivalent, but why? Is it related to Grothendieck's duality? Did I miss something obvious?



About a non-obvious (?) link between the jacobians of curves and differentials
Since you are in char zero, you can assume the ground field is the complex numbers. The inclusion of jacobians follows from the inclusion of spaces of differentials via the description in terms of periods and calculus.


