Is it grammatically correct to say:
He is our chemistry teacher. He is our English teacher.
etc? Can academic subjects be used as adjectives?
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Plenty of nouns can be used as adjectives; the grammatical term for this is noun adjunct, and it happens when a noun modifies another noun.
You can read all about them on the Wikipedia website (notice how I just used Wikipedia as an adjective there); other examples include: chicken soup, kitchen sink, light switch, brick house, biology teacher, and college professor.