A covariant functor $F$ is said to be left exact if whenever $$0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$$ is an exact sequence, then $$0 \to FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC$$ is also an exact sequence.

A covariant functor $F$ is said to be right exact if whenever $$A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$$ is an exact sequence, then $$FA \buildrel F\alpha \over \longrightarrow FB \buildrel F\beta \over \longrightarrow FC \to 0$$ is also an exact sequence.

A contravariant functor $F$ is said to be left exact if whenever $$A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C \to 0$$ is an exact sequence, then $$0 \to FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA$$ is also an exact sequence.

A contravariant functor $F$ is said to be right exact if whenever $$0 \to A \buildrel \alpha \over \longrightarrow B \buildrel \beta \over \longrightarrow C$$ is an exact sequence, then $$FC \buildrel F\beta \over \longrightarrow FB \buildrel F\alpha \over \longrightarrow FA \to 0$$ is also an exact sequence.

A (covariant or contravariant) functor is said to be exact if it is both left exact and right exact.