Grothendieck category

Remarks

Clauses 1 and 2 together form Grothendieck's axiom AB5. Every Grothendieck category has enough injective objects, and it also has all small limits.

Example: for a ring $R$, the category of left $R$-modules is a Grothendieck category. It has all colimits, and a filtered colimit of short exact sequences of modules is exact. The module $R$ is a generator: if $f \ne g \colon X \to Y$, pick $x$ with $f(x) \ne g(x)$ and let $h \colon R \to X$ be $r \mapsto r x$.

Non-example: the opposite of the category of abelian groups is abelian, has all small colimits, and has the generator $\Q/\Z$, but it fails clause 2. Filtered colimits in it are cofiltered limits of abelian groups, and these are not exact. For a prime $p$, take the inverse system of short exact sequences $0 \to p^n \Z \to \Z \to \Z / p^n \Z \to 0$ for $n \ge 1$. Its limit is $0 \to 0 \to \Z \to \Z_p \to 0$, where $\Z_p$ is the group of $p$-adic integers, and the map $\Z \to \Z_p$ is not surjective.