I'd like to find a reference to the following statement.

Let $k$ be a separably closed field of characteristic $p$, and $\phi: C_1\to C_2$ be a morphism of smooth projective curves over $k$. Assume that $\deg \phi$ is coprime to $p$. Let $\phi^*: J_2\to J_1$ be the morphism of the Jacobian varieties induced by the Picard functoriality, and $K$ be the kernel of $\phi^*$. Then the $\mathbb{G}_m$-dual $K^\vee$ of $K$ is isomorphic to covering group of the maximal abelian unramified covering of $C_2$ which is intermediate to $\phi: C_1\to C_2$.

This result (at least in characteristic 0) is probably well known to the experts, since it is used without any reference in a paper. Hopefully, I have formulated it correctly for characteristic $p$. I'd like to use it in my own paper, but I don't know where to find it in the literature.

Thank you.