lr adjoints imply lr exactness
If the functor $F$ is left adjoint to $G$, then $F$ is right exact and $G$ is left exact.
This is a corollary of the stronger result that left adjoints preserve colimits and right adjoints preserve limits ( here).
If the functor $F$ is left adjoint to $G$, then $F$ is right exact and $G$ is left exact.
This is a corollary of the stronger result that left adjoints preserve colimits and right adjoints preserve limits ( here).