Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional representations of $G$. Let $F\colon \operatorname{Rep}(G) \rightarrow \operatorname{Vect}_{\mathbb{C}}$ be the forgetfull (fiber) functor. According to the Tannkian formalism, group $G$ coincides with the group of tensor authomorphisms of the functor $F$.
Is it true that the Lie algebra $\mathfrak{g}$ is nothing else but the Lie algebra of endomorphisms $E\colon F \rightarrow F $ such that for any two representations $V,W \in \operatorname{Rep}(G)$ the following map $ F(V) \otimes F(W) \xrightarrow{E(V)\otimes \operatorname{Id} + \operatorname{Id} \otimes E(W)} F(V)\otimes F(W) $ coincides with the map $F(V\otimes W) \xrightarrow{E(V\otimes W)} F(V\otimes W)$ with respect to the identification $F(V)\otimes F(W) \simeq F(V\otimes W)$?
