TITLE: Exercise in Spanier to get universal coefficient theorem for cohomology with local coefficients
QUESTION [9 upvotes]: It is, it seems, rather well-known that there's no universal coefficient theorem for cohomology with local coefficients. However in Spanier's book Algebraic topology, there is an exercise that asks the reader to do something that at least goes by that name! Specifically (from page 283),

4 If $\Gamma$ is a local system of $R$-modules on $X$ and $G$ is an $R$-module, there is a local system $\mathrm{Hom}(\Gamma,G)$ of $R$-modules on $X$ which assigns to $x\in X$ the module $\mathrm{Hom}(\Gamma(x),G)$. Prove that
$$
\Delta^*(X,A;\mathrm{Hom}(\Gamma,G)) \approx \mathrm{Hom}(\Delta(X,A;\Gamma),G) 
$$
Deduce a universal-coefficient formula for cohomology with local coefficients.

It does not appear that he is assuming $R$ is a PID or anything special. A 2018 paper (Local Coefficients Revisited) says "there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited".
Though there's no actual statement in Spanier, as is apparent from the quote above!
What should the statement be, and why is it of limited application? (I'm not particularly interested in the proof at present, just what the theorem is meant to say!)

REPLY [2 votes]: After discussion with the OP, here's a little note that I wrote offering a modern point of view on this theorem - viewing it as a special case of the algebraic UCT, using the point of view that local coefficient systems are functors $X\to D(R)$, and that their co/homology can be interpreted as the homology of a limit/colimit of that functor.