  Separable Frobenius monoidal functors were defined and studied under that
name by Szlachanyi and by Day and Pastro, and in a more general context by
Cockett and Seely. Our purpose here is to develop their theory in a very
precise sense. We determine what kinds of equations in monoidal categories they
preserve. For example we show they preserve lax (meaning not necessarily
invertible) Yang-Baxter operators, weak Yang-Baxter operators in the sense of
Alonso Alvarez et al., and (in the braided case) weak bimonoids in the sense of
Pastro and Street. In fact, we characterize which monoidal expressions are
preserved (or rather, are stable under conjugation in a well-defined sense). We
show that every weak Yang-Baxter operator is the image of a genuine Yang-Baxter
operator under a separable Frobenius monoidal functor. Prebimonoidal functors
are also defined and discussed.
