Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is it true that in the category of $MA$-modules, $MA$ is dualizable? I want something weaker, actually. I want to justify that in the category of $MA$-modules and for smooth $k$-scheme $X$, we can 'dualize' $\text{Map}(MA, MA\wedge X_+)$ in the category of $MA$-modules to get $\text{Map}(MA\wedge X_+, MA)$. I don't want to pass to motivic cohomology. Am I missing something trivial?

Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is it true that in the category of $MA$-modules, $MA$ is dualizable? I want something weaker, actually. I want to justify that in the category of $MA$-modules and for smooth $k$-scheme $X$, we can 'dualize' $\text{Map}(MA, MA\wedge X_+)$ in the category of $MA$-modules to get $\text{Map}(MA\wedge X_+, MA)$. I want the process applied twice give back the original object. Am I missing something trivial?