Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication

$\Delta : \mathrm{S}\left(V\right) \to \mathrm{S}\left(V\right) \otimes \mathrm{S}\left(V\right)$

by

$\Delta\left(v_1v_2...v_n\right) = \sum\limits_{i=0}^n \sum\limits_{\sigma\in\mathrm{Sh}\left(i,n-i\right)} \left(v_{\sigma\left(1\right)}v_{\sigma\left(2\right)}...v_{\sigma\left(i\right)}\right) \otimes \left(v_{\sigma\left(i+1\right)}v_{\sigma\left(i+2\right)}...v_{\sigma\left(n\right)}\right)$,

where $\mathrm{Sh}\left(i,n-i\right)$ denotes the set $\left\lbrace \sigma\in S_n \mid \sigma\left(1\right) < \sigma\left(2\right) < ... < \sigma\left(i\right) \text{ and }\sigma\left(i+1\right) < \sigma\left(i+2\right) < ... < \sigma\left(n\right) \right\rbrace$ of all $\left(i,n-i\right)$-shuffles. The counit of this Hopf algebra is simply the projection from $S\left(V\right)$ onto $k$.

Definition. Let $k$ be a commutative ring, and $C$ be a $k$-coalgebra. A coderivation of $C$ means a $k$-linear map $c:C\to C$ such that $\Delta \circ c = \left(c\otimes\mathrm{id} + \mathrm{id}\otimes c\right)\circ \Delta$.

Remark. Coderivations behave, in some sense, dually to derivations (which is not surprising since the condition $\Delta \circ c = \left(c\otimes\mathrm{id} + \mathrm{id}\otimes c\right)\circ \Delta$ is a kind of dual to the Leibniz identity, when the latter is written in pointfree notation): First of all, if $c:C\to C$ is a coderivation, then $c^{\ast} : C^{\ast}\to C^{\ast}$ is a derivation. The converse holds at least when $C$ is finite-dimensional. As an exercise in reversing arrows, the reader can prove that $\varepsilon\circ c=0$ for every coderivation $c$ of a coalgebra (in analogy to the equality $d\left(1\right)=0$ which holds for every derivation $d$ of an algebra).

Theorem. Let $k$ be a field of characteristic $0$. Let $V$ be a $k$-vector space. Then, the maps

$\mathrm{Hom}\left(S\left(V\right),V\right) \to \mathrm{Coder}\left(S\left(V\right)\right)$,

$X\mapsto \mu \circ \left(\mathrm{id}\otimes X\right) \circ \Delta$ (where $\mu$ is the multiplication morphism $S\left(V\right)\otimes S\left(V\right)\to S\left(V\right)$)

and

$\mathrm{Coder}\left(S\left(V\right)\right) \to \mathrm{Hom}\left(S\left(V\right),V\right)$,

$c\mapsto \pi_1\circ c$ (where $\pi_1$ is the projection from $\mathrm{S}\left(V\right)=\bigoplus\limits_{i\in\mathbb N}\mathrm{S}^i\left(V\right)$ onto the addend $\mathrm{S}^1\left(V\right)=V$)

are mutually inverse isomorphisms.

This is how I understand Chapter 5, Theorem 4.19 in Eckhard Meinrenken, Clifford algebras and Lie groups. (Unfortunately, the statement of the theorem in Meinrenken's text is obscured by the fact that one direction of the isomorphism - the easy one - is not written out explicitly.) The proof given in this text uses the characteristic-$0$ assumption: first, by assuming "WLOG" that a generic element of $\mathrm{S}\left(V\right)$ has the form $e^v$ for some $v\in V$ (this is made formal by going over to formal power series, but stripped of this formality, this is exactly what is known as umbral calculus for over a century), and second, by using the fact that the primitives of $\mathrm{S}\left(V\right)$ all come from $V$.

Question. Does the above theorem hold in arbitrary characteristic?

I am sure this is intimately related to the question whether $\mathrm{S}\left(V\right)$ is the cofree graded cocommutative coalgebra over $V$ (or at least whether it is "cogenerated" in degree $1$, whatever this means precisely!). Unfortunately, the only case when I am sure of this is the characteristic-$0$ case, so this is of no help to me. Loday-Valette does not seem to care for positive characteristic too much, either, and it is difficult for me to find any other source.