Remark 9.2.5.3. In the situation of Example 9.2.5.2, we can replace $(3)$ by the following a priori weaker condition:
- $(3')$
For every short exact sequence
\[ 0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z \rightarrow 0 \]in the abelian category $\operatorname{\mathcal{C}}$, the sequence
\[ 0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z) \]is also exact.
The implication $(3) \Rightarrow (3')$ is immediate. Conversely, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an additive functor which satisfies condition $(3')$, and suppose we are given an exact sequence $0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z$ in $\operatorname{\mathcal{C}}$; we wish to show that the sequence $0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z)$ is exact in $\operatorname{\mathcal{D}}$. Applying condition $(3')$ to the short exact sequence
we deduce that $F(u): F(X) \rightarrow F(Y)$ is a monomorphism, whose image is the kernel of the map of the map $F( Y ) \rightarrow F( \operatorname{coker}(u) )$. To show that this agrees with the kernel of $F(v)$, it will suffice to show that $v$ induces a monomorphism $F(\operatorname{coker}(u)) \rightarrow F(Z)$ in the abelian category $\operatorname{\mathcal{D}}$. This follows by applying condition $(3')$ to the short exact sequence