Deligne-Knop tensor categories and functoriality

A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},\delta )$ from any regular category $\mathcal{A}$ and a fixed degree function $\delta$. A special case of this construction are the Deligne categories $\underset{}{\mathrm{Rep}}(S_t)$ and $\underset{}{\mathrm{Rep}}(G{L}_t({\mathbb{F}}_q))$. We discuss when a functor $F:\mathcal{A}\to {\mathcal{A}}^{\mathrm{\prime }}$ between regular categories induces a symmetric monoidal functor $\mathcal{T}(\mathcal{A},\delta )\to \mathcal{T}({\mathcal{A}}^{\mathrm{\prime }},{\delta }^{\mathrm{\prime }})$. We then give a criterion when a pair of adjoint functors between two regular categories $\mathcal{A},{\mathcal{A}}^{\mathrm{\prime }}$ lifts to a pair of adjoint functors between $\mathcal{T}(\mathcal{A},\delta )$ and $\mathcal{T}({\mathcal{A}}^{\mathrm{\prime }},{\delta }^{\mathrm{\prime }})$.