It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That is to say that there is not a single spectrum $X$ in the category of harmonic spectra such that $[X,\bigvee X_i]=\bigoplus[X,X_i]$. Take a finite spectrum in this category (for instance the sphere spectrum $\mathbb{S}$). Does anyone have an example of a coproduct in the category of harmonic spectra for which the above is NOT true? While Strickland's proof makes sense, it seems completely amazing to me. What HAPPENS to the sphere and the cells of finite spectra when we localize? It seems that if something were small to begin with, i.e. the above homotopy groups factored globally, killing some spectra shouldn't change this! Note that the same question can be asked of the $BP$-local category.

Are there conditions (other than finiteness) that we can put on the coproduct to make the statement true when the object on the left is finite (i.e. in the thick subcategory generated by the sphere)?