I'm reading Milne's book of algebraic geometry and he gives the following criterion for direct limits:
An $R$-module $M$ together with $R$-linear maps $\alpha^i: M_i \to M$ is the direct limit of a system $(M_i,\alpha^i_j)$ if and only if
$i)$ $M = \cup_{i \in I} \alpha^i (M_i)$
$ii)$ $m_i \in M_i$ maps to zero in $M$ if and only if it maps to zero in $M_j$ for some $j \geqslant i$, and
$iii)$ $\alpha^i = \alpha^j \circ \alpha^i_j$ for all $j \geqslant i$.
I could prove some parts until now, but not all.