Let $k$ be a field (more genererally a commutative ring) and consider the category of vector spaces (modules). The double dual endofunctor has a structure of monad where
the unit is $$ \begin{align} \eta_V \colon V &\to V^{\vee \vee} \\ v &\mapsto (\alpha \mapsto \alpha(v)), \end{align} $$
and the multiplication is $$ \begin{align} \mu_V \colon \underbrace{((((V \to k) \to k) \to k) \to k)}_{V^{\vee \vee \vee \vee}} &\to \underbrace{(V \to k) \to k}_{V^{\vee \vee}} \\ \psi &\mapsto (\alpha \mapsto \psi(\eta_{V^{\vee}}(\alpha))). \end{align} $$
Question: what is an Eilenberg-Moore algebra for this monad?
Going by definition, it is a pair $(V, \xi \colon V^{\vee \vee} \to V)$ where the structural morphism $\xi$ is compatible with unit and multiplication.
Compatibility with unit says that $$ \xi(\alpha \mapsto \alpha(v)) = v $$ for every $v \in V$.
I have trouble with compatibility with multiplication and in giving a description for the final structure.