Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces.
Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category $\mathcal{M}^{*}:=Hom_{vec}(\mathcal{M},vec)$, with action given by precomposition (generalising the dual representation of a finite group), such that $\mathcal{M}\boxtimes\mathcal{M}^*$ is a $vec_{G}$-module category under the diagonal action of $vec_{G}$ and define an adjoint pair of $vec_G-$module functors:
$coev_{\mathcal{M}}:vec\rightarrow \mathcal{M}\boxtimes\mathcal{M}^*$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbb{C}\mapsto \oplus_{i}M_{i}\boxtimes M^{i}$
$ev_{\mathcal{M}}: \mathcal{M}\boxtimes\mathcal{M}^* \rightarrow vec$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, M\boxtimes M^{*}\mapsto M^*(M)$
where $vec$ above is considered as the $vec_{G}$-module category defined by the forgetful functor $F:vec_{G}\rightarrow vec$ and $\{X^{i}\}$ is a set of simple objects of $\mathcal{M}$ and $\{ M^{i} \}$ is a set of simple objects of $\mathcal{M}^*$ such that $M^{j}(M_{i})=\mathbb{C}$ for $i=j$ and the zero vector space else.
My question is, considering the composition:
$coev\circ ev:vec\rightarrow \mathcal{M}\boxtimes\mathcal{M}^*\rightarrow vec\in End_{Vect_{G}}(vec)\simeq Rep(G)$.
Is there a way to find a $vec_{G}$ module category $\mathcal{M}$ such that the product $coev\circ ev$ defines a given representation $(\rho,V)$ of $G$? Additionally can all representations of $G$ be found by utilising the above composition of module functors?
Im also interested in the case of $vec^{\alpha}_{G}$ where $\alpha\in H^{3}[G,U(1)]$ defines a non-trivial monoidal associator for $vec_{G}$ if anyone understands the generalisation.
