Dioperads, Frobenius monoidal functors and duality

Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a $d$-duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object $d$. For any enriched dioperad $\mathcal{P}$, we define a $d$-twist $\mathcal{P}\{d\}$ and prove that, in a $d$-duality context, the right adjoint sends $\mathcal{P}$-algebras to $\mathcal{P}\{-d\}$-algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an $\infty$-categorical extension of the theory.