Let $ \text{Vect} $ be the category of finite dimensional vector spaces over an algebraically closed field. The idempotent completion of a $\text{Vect}$-category $ \mathcal{C} $ may be though of in two different ways:

• As a category with objects $ (X,e_X) $ where $ X $ is in $ \mathcal{C} $ and $ e_X $ is an idempotent in $ \text{End}(X) $.
• As the the additive subcategory of $ \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $ (linear functors from $\mathcal{C}^{\text{op}} $ to $ \text{Vect}$) generated by direct summands of $ \mathbb{Y}(X) $ for $ X $ in $ \mathcal{C} $ (where $ \mathbb{Y} $ is the Yoneda embedding).

To an object-idempotent pair $ (X,e_X) $ one can associate the (contravariant) functor

$$ \begin{align} (X,e_X)^\sharp \colon \mathcal{C} &\to \text{Vect} \\ Y &\mapsto \{ f \in \text{Hom}(Y,X) \mid e_X \circ f = f \} \end{align} $$

which provides an equivalence between these two categories.

Question

From the above discussion we see that the following are equivalent

• $ \mathbb{Y}\colon \mathcal{C} \to \text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect}) $ is an idempotent completion.
• Every contravariant functor from $\mathcal{C}$ to $\text{Vect}$ is naturally isomorphic to $ (X,e_X)^\sharp $ for some $X$ in $\mathcal{C}$ and some idempotent $e_X$ in $\text{End}(\mathcal{C}) $.

Under what assumption on $ \mathcal{C} $ do these statements hold? For example, they hold if $ \mathcal{C} $ is semisimple with finitely many simple objects. I would like to know if they still hold without the assumption that $ \mathcal{C} $ has finitely many simple objects. I would also like to know if they hold when $\text{Fun}(\mathcal{C}^{\text{op}}, \text{Vect})$ is semisimple (even if $ \mathcal{C} $ isn't).