Final answer:
Fungal cells are eukaryotic because they possess a membrane-bound nucleus and organelles, and have cell walls made of chitin.
Step-by-step explanation:
The cells that make up fungi are indeed eukaryotic. This is clear from the fact that fungal cells have a complex cellular organization with a membrane-bound nucleus and membrane-bound organelles such as mitochondria, endoplasmic reticulum, and Golgi apparatus. In contrast to prokaryotic cells, which lack a defined nucleus and these complex organelles, the eukaryotic cells of fungi also have unique cell walls made of chitin rather than cellulose.
These characteristics firmly place fungi within the domain Eukarya.