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.)