 Milnor exact sequence for maps of spectra

For $D={\mathrm{hocolim}}_j{T}_j$, applying $[-,E{]}_{\mathrm{st}}$ to the telescope cofiber sequence of spectra givesHere ${\lim }^1$ is the cokernel of $1-r$ on the product of the inverse tower, where $r$ uses its bonding maps. This records the obstruction to uniqueness when compatible stage maps are realized.