Suppose I have an $H$-space $H$ and a topological group $G$, such that for *compact* spaces $X$ there is a natural equivalence of group valued functors

$[X, H] \to [X, G]$

Now $H$ is a (non-finite) CW-complex (in my case it is $BU_{\otimes}$), but $G$ is some really huge not even locally compact space. Can I somehow deduce from this, that $H$ and $G$ are weakly homotopy equivalent?

This would of course follow, if I had a map $H \to G$ inducing the natural equivalence, right? But I only have the natural transformation above. Can I deduce the existence of a map $H \to G$ from this somehow?