$X$ 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!