Categorical description of adjoint representation My apologies for a slightly vague question here. Let us fix a Lie algebra $\mathfrak{g}$ over a field $k$. Consider the symmetric monoidal category $\operatorname{Rep}_{k}^{\otimes}(\mathfrak{g})$ of representations of $\mathfrak{g}$. If you wish, you may consider variants of this with dg-categories, stable $(\infty, 1)$-categories etc. There is an element $\mathfrak{g}^{ad}$ of this category, moreover it is a Lie algebra object with respect to the monoidal product $\otimes$. Is there some way to characterize this representation categorically? Feel free to add additional structure so that the answer is positive, basically I'd like to know if there is a sensible axiomatization of "(symmetric monoidal) category with adjoint object".
 A: For reference, here is the answer suggested by Adrien above.
Let $(C, \otimes) $ be an abelian complete and co-complete Cartesian-closed symmetric monoidal category over a field $k$. We write $Map$ for internal mapping objects and $\mathbb{1}$ for the unit.
An augmented algebra in $(C, \otimes) $ is an associative unital algebra object, $X$ together with a map of algebras $\eta:X\rightarrow\mathbb{1}$. We define the cotangent object of $(X, \eta)$ as $\Omega_{\eta}X:= ker(\eta) /ker(\eta) ^{2}$.
We now define the adjoint object of $C$ as the cotangent object of the categorical end, $\int_{C} Map$, at its natural augmentation. (The augmentation is induced from the map $\int_{C} Map\rightarrow End(\mathbb{1})\cong\mathbb{1}$).
In the case of lie algebras the end is $(U\mathfrak{g})^{ad}$ and so we have $$\mathfrak{g}^{ad} = \Omega_{\eta}\int_{Rep(\mathfrak{g})} Map$$.
Edit. This is incorrect, as pointed out in the comments. Adrien's comments above still suffice to produce $U\mathfrak{g}^{ad}$ categorically, however the attempt in this post to leverage this to a construction of $\mathfrak{g}^{ad}$ is flawed.
