homotopy direct limits $X$ is said to be the homotopy direct limit of the sequence of subsets $X_1\subset X_2\subset ...$ if the projection $\cup_i X_i\times [i,i+1] \rightarrow X$ is a homotopy equivalence.  
The following is true:
Suppose $X, Y$ are homotopy direct limits of the sequences $X_1\subset X_2 \subset ...$ and $Y_1 \subset Y_2 \subset ...$ respectively. Then if $f: X\rightarrow Y$ and each $f|_{X_i}: X_i \rightarrow Y_i$ is a homotopy equivalence then $f$ itself is a homotopy equivalence.  
My Question:  Does anyone know off the top of his or her head whether the map $f$ can be replaced by a sequence of homotopy equivalences $f_i: X_i \rightarrow Y_i$ where the resulting diagram is homotopy commutative (i.e. $(Y_i\subset Y_{i+1})\circ f_i$ is always homotopic to $f_{i+1}\circ (X_i\subset X_{i+1})$)? I.e. are the spaces still homotopy equivalent? Thanks!
 A: Yes, the spaces are still homotopy equivalent.
Your situation with homotopy direct limits is a special case of a homotopy colimit and the defining property of homotopy colimits is exactly what you're asking for. See, for example, Theorem 1.4 of Rainier M. Vogt's "Homotopy limits and colimits". Be warned, there is a lot to read and to prove in order to get to this result.
However, since you are interested only in directed systems we should be able to find a more direct (no pun intended) proof. At our disposal we have the homotopy equivalences between $X_i$ and $Y_i$ and the homotopies assuring associativity of our diagram. With these tools can you see a way of constructing a natural-looking continuous map between $\cup_i X_i\times [i,i+1]$ and $\cup_i Y_i\times [i,i+1]$? Can you construct a homotopy inverse?
I did the former part, but I failed to find the homotopy inverse. I hope you'll have more luck than me, though!
EDIT: See the comments for a discussion on how such a homotopy inverse might be constructed.
