  Given a tensor functor between tensor categories $\mathcal{C}$ and
$\mathcal{D}$, we give criteria that, under certain assumptions, the
Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of
the other one. We also give an affirmative answer to Natale's question asking
if the class of Frobenius tensor categories is closed under exact sequences.
