Let $X$ be a CW-spectrum. It is well-known that $[\\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely, $H$ has this form).


What about $[X, \\_]$? Is it a homology theory? (I do not claim every homology is corepresented by a spectrum.)



Is $[X, \_]$ a homology theory?
This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a homology theory from a spectrum $X$ is to consider $\pi\_\*(X\otimes -)$, note that for compact $X$ your $[X,-]$ is of this form as well, since
$$
[X,-] = \pi\_\*(DX\otimes -)
$$
by Spanier-Whitehead duality.


