I am currently reading this paper on matroids.
http://arxiv.org/pdf/1411.2277.pdf
I am reading the proof to proposition 3.17 which involves transversal matroids.
"In particular, any coloop is always adjacent to every vertex in W. So without loss of generality, we assume that M is coloop-free."
Here, the transversal matroid M can be presented by a left locally finite bipartite graph (V,W,E). I understand that since the transversal matroid is finitary, any vertex must only have finitely many edges. If there is a coloop, then it's adjacent to every vertex W and thus W is finite. I'm not sure if I understand why proving proposition 3.17 in the coloop free case proves it in the most general case. Is the argument as follows? Let M be a finitary transversal matroid. Let S be the set of coloops of M. Then M-S has a unique maximal presentation. To include S, just add the missing vertices of V and connect them to every edge of W. Is my above understanding correct?