= Milnor exact sequence for maps of spectra
{c}
For $D=\operatorname{hocolim}_jT_j$, applying $[-,E]_{\mathrm{st}}$ to the telescope <cofiber sequence of spectra> gives
$$
0\longrightarrow\lim\nolimits^1_j[\Sigma T_j,E]_{\mathrm{st}}\longrightarrow[D,E]_{\mathrm{st}}\longrightarrow\lim_j[T_j,E]_{\mathrm{st}}\longrightarrow0.
$$
Here $\lim^1$ 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.
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