Homotopy of localisations of colimits Let $X_k$ be a family of spectra equipped with maps $f_k: X_k \to X_{k+1}$.  If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy colimit $X_\infty := hocolim \; X_k$ of the sequence by
$$[Y, X_\infty] = \lim_{k \to \infty} [Y, X_k]$$
Now if I assume that the $X_k$ are local for Morava K-theory $K(n)$, then it need not be the case that the homotopy colimit is, too.  In order to force this to be true, I can simply re-localise $X_\infty$.
My question is: is there a similar formula for $[Y, L_{K(n)} X_\infty]$? Does it help that the $X_k$ were local to begin with?  I'm happy to assume that the $X_k$ have finite $K(n)_*$, if that makes life any easier.
 A: There are two natural finiteness conditions that you might impose on $Y$.  The stronger one says that the Morava $E$-theory of $Y$ is finite in each degree, or equivalently that $Y$ is a retract of $L_{K(n)}Y_0$ for some finite spectrum $Y_0$ with $E(n-1)_*Y_0=0$.  If this holds, then $[Y,L_{K(n)}X_\infty]$ is the colimit of the terms $[Y,X_k]$, the key point being that $DY\wedge X_\infty$ is already $K(n)$-local (even though $X_\infty$ itself is not).
If $Y$ does not have the above smallness property then it will rarely be true that $[Y,L_{K(n)}X_\infty]$ is the colimit of $[Y,X_k]$.  For the canonical counterexample, let $\{M_k\}$ be a tower of generalised Moore spectra of the usual kind, so 
$$ BP_*(M_k)=BP_*/(v_0^{a_0},\dotsc,v_{n-1}^{a_{n-1}}) $$
with the exponents $a_i$ tending to infinity as $k$ increases.  Put $Y=L_{K(n)}S^0$ and $X_k=F(M_k,L_{K(n)}S^0)$.  In this case it is known that $X_\infty=L_{K(n)}S^0$ so $[Y,X_\infty]$ contains $\mathbb{Z}_p$, but each group $[Y,X_k]$ is finite so the colimit is torsion.
