I have the following exercise:
Show that for a semisimple, additive category $\mathcal{C}$ the following holds: the structure of a k-linear category over $\mathcal{C}$ is the same as $\mathcal{C}$ being a module category over $\mathrm{vect}_k$, k a field.
Now I have come so far, that I defined the following:
For $\mathcal{C}$ being a module category I want to show that $\mathrm{Hom}(C, C') \forall C \in \mathcal{C}$ is a vector space. Since $\mathcal{C}$ is additive this leaves me with scalar multiplication. But I got the action on morphisms which means that for $f: k \rightarrow k$ a vector morphism and $\mathcal{C} \ni g: C \rightarrow C'$ I get: $f \otimes g: k \otimes C \rightarrow k \otimes C'$.
With $\mathrm{End}(k, k) \cong k$ I get the scalar multiplication.
The other way around: Let $\mathcal{C}$ be a $k$-linear category. Then I can define $V.C$ as the object representing $V \otimes_k \mathrm{Hom}(-, C)$, where $\otimes_k$ is the tensor product of vector spaces. For morphisms $\mathrm{vect} \ni f: V \rightarrow V'$ and $\mathcal{C} \ni \phi: C \rightarrow C'$ it follows that \begin{align*} f \otimes \phi \in &\mathrm{Hom}_{\mathcal{C}}(V \otimes C, V' \otimes C') \\ &\cong V' \otimes_k \mathrm{Hom}_{\mathcal{C}}(V \otimes C, C') \\ &\cong V' \otimes_k V^* \otimes_k \mathrm{Hom}_{\mathcal{C}}(C, C') \\ &\cong \mathrm{Hom}_{\mathrm{vect}}(V, V') \otimes_k \mathrm{Hom}_{\mathcal{C}}(C, C') \end{align*}
But what I have left to show is that those constructions are kind of "inverse" to each other. and that is where I am lost.
How can I show that with these two constructions the two structures are equivalent?
Thanks in advance!