This is not a simple process.. If your text does not offer a good description. As good place to start reading is under Characteristics of Teleomorphs found in this reference. Aspergillus has a teleomorphic (sexual) as well as anamorphic (asexual) life cycle, both together comprise the whole life form or holomorph.
I'll attempt to condense the concept, in the following:
The teleomorphs exhibit a unique nuclear phenomenon called the dikaryon. After sexually compatible nuclei from different mycelia, have been brought together by anastomosis (fusing of two hyphae), these nuclei pair off, but don't fuse immediately to form a diploid (2N) zygote. Instead, they go on dividing to form what are called dikaryotic hyphae, in which each compartment (cell) has two sexually compatible haploid (1N) nuclei. These 2 haploid nuclei will eventually fuse after further development, into diploid (2N) nuclei which is followed by meiosis where the genome finally has the chance to be reshuffled (just like meiosis in ordinary sexual reproduction).
glossary: hypha is threadlike fungal filament, (plural - hyphae). mycelium is a mass of hyphae (plural - mycelium)
glossary: mycelium is a mass of hyphae (plural - mycelium)
sorry for my typo, plural of mycelium is mycelia
While I'm on typos, it appears you made one in the spelling of this organism. The actual spelling of this genus is Aspergillus and it includes over 185 species. The teleomorphic stage has only been observed in some of these species. That doesn't mean they don't exist, just that no-one has found evidence yet).