I have two sources, which claim that the category of perverse sheaves on $\mathbb C$ constructible with respect to the stratification $0$ and $\mathbb C^*$ is equivalent to the category of certain representations of a quiver. In both cases the quiver consists of two dots with one arrow $u,v$ in each direction between them.
The first source considers such representation such that both 1+uv and 1+vu are invertible.
The second source only wants 1+uv to be invertible.
Now I have to admit that I understand neither of the proofs completely so my question is which description is correct?