But the terms haploid and diploid don't really capture the special nature of fungal genetics, so of course mycologists have invented a special set of terms for that. You see, even after diploidization, the nuclei of the two partners don't fuse together into one big nucleus, the way they do in plants and animals. Instead, each cell has two separate nuclei that pretty much function independently of one another. In fact, certain coenocyticfungi can incorporate other individuals repeated, and routinely end up with the nuclei from half a dozen or more different bionts roaming around inside their mycelium. Each nucleus makes its own contribution - - one study has shown that as low as 4% of nuclei that are resistant to a fungicide can confer resistance on the whole mycelium.
The special terms for this special arrangement are as follows: the original, " haploid" mycelium is called monokaryotic and the diploidized version is called dikaryotic (from the Greek karyon, meaning "kernel"). Presumably a multi-member coenocytic mycelium would be polykaryotic. At the time of Snell & Dick(1957) , the spellings dicaryon and monocaryon were preferred (by them, at least), but since then the versions with a "k" in the middle seem to have won out.
The only time that fungal nuclei ever actually merge with their cell-mate is just before meiosis and spore production, and this process is called karyogamy (again, caryogamy used to be the preferred spelling). The two nuclei duplicate themselves, fuse, mix-and-match their genes, divide, and then divide again, producing four new, sexually shuffled nuclei which become the nuclei of the four spores produced on a "typical" basidium. I haven't managed to find an account of what goes on differently in bisterigmatefungi, the eight spores in a typical ascus, the five-to-seven spored basidia of the Cantharellaceae, or the irregular, sometimes huge numbers of spores per ascus that certain ascomycetes produce. But then, I haven't looked terribly hard.