Say we have a Lie group $G$. The category of (finite, complex) representations $\mathsf{Rep}\,G$ contains the adjoint representation $\mathfrak{g}$ which has many special properties. For instance $\mathfrak{g}$ is a Lie algebra internal to the category. Furthermore, there is a natural transformation $$\eta: (\mathfrak{g}\otimes -)\rightarrow\text{id}_{\mathsf{Rep}\,G}$$ which satisfies $$\eta_{\mathbf a\otimes\mathbf b} = \eta_{\mathbf a}\otimes 1_{\mathbf b}+(1_{\mathbf a}\otimes\eta_{\bf b})\circ(\beta_{\mathfrak g,\mathbf a}\otimes 1_{\mathbf b})$$ where $\beta$ is the braiding, which gives the action of the adjoint on a representation.
I am interested in the generalization of the adjoint representation to abstract symmetric tensor categories. Have such categories been considered in the literature, and if so where can I find out more?
