An inspectorate is the official group of inspectors — the body of people whose job is to inspect schools, factories, prisons, and so on. It names the organisation, not one person walking around with a clipboard. You will hear it especially in British and formal English: the schools inspectorate, a factory inspectorate, the nuclear inspectorate. American English more often says inspectors, an inspection agency, or a board. Though actually, newspapers sometimes use inspectorate as if it were just a fancy word for inspection, but the -ate here is the office and the people in it, the way a directorate is the body of directors. In Taiwan you already know 督學 and 勞動檢查; this English word is the whole official team and the office around them.