What Separable Frobenius Monoidal Functors Preserve

Abstract. Separable Frobenius monoidal functors were de ned and studied under that name in [10], [11] and [4] and in a more general context in [3]. 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 [1], and (in the braided case) weak bimonoids in the sense of [8]. In fact, we characterize which monoidal expressions are preserved (or rather, are stable under conjugation in a well-de ned 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 de ned and discussed.