Suppose, for each $n>0$, I have two complexes of abelian groups $(A\_{n}, d)$ and $(B\_{n},d')$
and a quasi-isomorphism 
$$f\_{n}: A\_{n} \rightarrow B\_{n}.$$
Furthermore, suppose I have maps of complexes $A\_1 \rightarrow A\_2 \rightarrow A\_3 \dots$
and similarly for $B\_n$, compatible with $f\_n$.

Is it true that the map of direct limit complexes
$$\lim\_{n} f\_{n}: \lim A \rightarrow \lim B$$
is also a quasi-isomorphism?

Complexes can mean unbounded or maybe just bounded above/below, depending on your preference.

Also: what if I replace the limit over natural numbers with a more general filtered colimit?