I give Coprinellus simply as a reasonable guess as to the genus. Two or three decades ago small mushrooms such as these would have been placed in Coprinus. Molecular studies split the genus, with a few species staying in Coprinus but the bulk of them being moved to Coprinellus, Coprinopsis or Parasola. In those three the caps have radial grooves and are thin (in general, very thin in Parasola, with the caps often flat or very shallowly curved, much like a dainty parasol, hence the name). In Coprinellus and Coprinopsis the caps often don't flatten out as much. You'll have guessed that since I talk of 'in general' and 'mostly', macroscopic features won't always tell you for certain which genus you have. Correct. One final point: in these genera the mature spores are black so the gills of the mature mushrooms are very dark. There are also pale-spored genera, with thin-fleshed, radially-grooved caps.