Variant 9.2.5.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is right exact if it preserves finite colimits: that is, if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is left exact. When $F$ is an additive functor between abelian categories, this condition can be restated as follows:
- $(\ast )$
For every exact sequence
\[ X \xrightarrow {u} Y \xrightarrow {v} Z \rightarrow \]in the abelian category $\operatorname{\mathcal{C}}$, the sequence
\[ F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z) \rightarrow 0 \]is exact in the abelian category $\operatorname{\mathcal{D}}$.
Moreover, it suffices to verify condition $(\ast )$ in the special case where $u$ is an monomorphism (Remark 9.2.5.3).