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Suppose $$F:{\mathcal {D}}(X)\rightarrow {\mathcal {T}}$$ F : D ( X ) → T is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety X to a triangulated category $${\mathcal {T}}$$ T . If F possesses left and right adjoints, then the Bondal–Orlov criterion gives a simple way of determining if F is fully faithful. We prove a natural extension of this theorem to the case when X is a smooth and proper DM stack with projective coarse moduli space.

Bondal–Orlov fully faithfulness criterion for Deligne–Mumford stacks