$\newcommand{\dR}{\mathrm{dR}}$**Edit**: I originally asked this question on MSE, but migrated it to MO after a long period of inactivity and a recommendation from another user.

Let $X$ be a complex elliptic curve and $e$ the identity element of $X$. Let $E^\times$ denote the punctured curve $X\backslash\left\{e\right\}$. In their 2007 paper *Towards Multiple Elliptic Polylogarithms*, Levin and Racinet compute the de Rham fundamental group $\pi_1 ^{\dR} (E^\times, e)$. To do this they prove that there is an equivalence of categories (Theorem 1)

$$R\mathsf{-NMod}\xrightarrow{\sim} \mathsf{NConn}(X;e)$$

where:

- $R = \mathbb{C}\langle\langle \mathbf{t}, A\rangle\rangle$ is the Hopf algebra of noncommutative power series in two formal variables $\mathbf{t}$ and $A$. $R$ is also the completed universal enveloping algebra of the free Lie algebra $\mathbb{L}(\mathbf{t}, A)$ generated by $\mathbf{t}$ and $A$. The coproduct in $R$ is given by

$$\Delta(\mathbf{t}) = \mathbf{t}\otimes 1 + 1\otimes\mathbf{t},\quad \Delta(A) = A\otimes 1 + 1\otimes A;$$

- $R$-$\mathsf{NMod}$ is the category of (finitely generated) "nilpotent left $R$-modules", which I take to mean (as they define) finite-dimensional complex vector spaces $V$ with two
**nilpotent**endomorphisms $\mathbf{t}, A\in\text{End}(V)$; - $\mathsf{NConn}(X;e)$ is the category of "nilpotent vector bundles" on $X$ with log-poles at $e$, which I take to mean
**unipotent**vector bundles i.e. those $\mathcal{V}$ possessing a finite filtration by subbundles (with compatible connections) such that the quotients are all isomorphic to the trivial bundle with trivial connection $(\mathcal{O}_X, d)$. This category is often called $\mathsf{Un} (X;e)$ or something to that effect (e.g. in the papers of M. Kim).

By definition the de Rham fundamental group $\pi_1^{\dR} (E^\times;b)$ is the Tannakian fundamental group of the category $\mathsf{Un}(X;e)$, with fiber functor $\omega$ sending a bundle to the stalk over a basepoint $b$ of $X$ (or for $e$, a suitable notion of "tangential basepoint"). By the above equivalence, it follows that there is an isomorphism between $\pi_1^{\dR} (E^\times;e)$ and the Tannakian fundamental group of $R$-$\mathsf{NMod}$ (with the forgetful functor as fiber functor).

In Corollary 2.2.7, Levin and Racinet claim that this Tannakian fundamental group is

$$\pi_1^{\dR} (E^\times; e)\cong \exp \mathbb{L}(\mathbf{t}, A).$$

I don't understand how this is calculated as the Tannakian fundamental group of $R$-$\mathsf{NMod}$. Would anyone be able to help me or point to how this is calculated? Many thanks!