Morita equivalence of acyclic categories (Crossposted to MathOverflow.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. Suppose that they are Morita equivalent in the sense that the abelian categories $\text{Fun}(C, \text{Vect})$ and $\text{Fun}(D, \text{Vect})$ are equivalent (where $\text{Vect}$ is the category of vector spaces over a field $k$, say algebraically closed of characteristic zero). Are $C, D$ equivalent? (If so, can we drop the finiteness condition?) 
Without the acyclic condition this is false; for example, if $G$ is a finite group regarded as a one-object category, $\text{Fun}(G, \text{Vect})$ is completely determined by the number of conjugacy classes of $G$, and it is easy to write down pairs of nonisomorphic finite groups with the same number of conjugacy classes (take, for example, any nonabelian group $G$ with $n < |G|$ conjugacy classes and $\mathbb{Z}/n\mathbb{Z}$). 
On the other hand, I believe this result is known to be true if $C, D$ are free categories on finite graphs by basic results in the representation theory of quivers, and I believe it's also known to be true if $C, D$ are finite posets. 

On MO, Benjamin Steinberg links to a paper of Leroux with a counterexample. 
