Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?
Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later discovered to be–who knew?–categorical duals of each other.
Would it then be entirely arbitrary whether to rename doohickeys as "codoodads" or rename doodads as "codoohickeys"?
The fact that, for me at least, cothingies are typically far more exotic than their duals suggests that there is no intrinsic "coquality" that distinguishes cothingies from thingies, and therefore, all other things being equal, it is the more exotic member of the pair that gets the coname. (Otherwise the extreme underrepresentation of cothingies in my prior mathematical education is hard to explain.) But occasionally I come across comments that suggest that some mathematicians at least have an intuitive sense of how to classify arbitrary mathematical entities as either thingies or cothingies. Therein the nundrum.