Most chams really like flies. It's hard for them to resist a fly flying around. I free range so I can't give mine flies but my female will zap her one outside sometimes.
There were a few nats flying around Ozzies habitat and I saw him catching them and eating them. I guess it is ok for him to eat them? Its getting really warm here and I guess the moisture from the live plants are drawing nats but they don't last that long in there when Oz sees them
Some times when i take him and his cage out side to get some natural UVB some times little nats work there way in and right when he sees the there gone.