Suppose having a set of direct systems of abelian groups
$\ldots\{G_{\alpha}\}_{\alpha\in A}$, $\{G_{\beta}\}_{\beta\in B}$, $\{G_{\gamma}\}_{\gamma\in \Gamma}\ldots$
If there is a (long) exact sequence for certain indexes:
$\cdots\longrightarrow G_\alpha\longrightarrow G_\beta\longrightarrow G_{\gamma}\longrightarrow\cdots\ $ Then, is the following sequence exact?
$\cdots\longrightarrow \varinjlim G_\alpha\longrightarrow \varinjlim G_\beta\longrightarrow\varinjlim G_{\gamma}\longrightarrow\cdots\ $