 Universal evaluation phantom map

Choose one representative of each finite spectrum and form $X={\bigvee }_{(F,u:F\to D)}F$. Let $e:X\to D$ evaluate each summand and let $p:D\to \mathrm{cofib}(e)$. Every map from a finite spectrum factors through its corresponding summand, so $p$ is a phantom map of spectra. If $p=0$, exactness supplies a section of $e$, making $D$ a retract of $X$. Consequently a spectrum that is not a retract of any wedge of finite spectra has a nonzero such phantom.