In the top answer to the question "Is there a map of spectra implementing the Thom isomorphism?" it is explained (with a reference to Rudyaks book) that from a rank $r$ vector bundle $\mu:V\to X$, a spectrum $E$ with multiplication $m:E\wedge E\to E$ and a Thom class $t:X^\mu\to \Sigma^r E$ one gets the Thom isomorphism:

The (cohomological) Thom isomorphism sends a map $x:X_+\to E^n$ of spectra to the composition $$ X^\mu\to X_+\wedge X^\mu\xrightarrow{x\wedge t}E^n\wedge E^r\xrightarrow{m}E^{n+r}. $$ The induced isomorphism on homotopy groups is then $E^{n}(X_+)\cong E^{n+r}(X^\mu)$.

Is it possible to give an inverse map, i.e. a map sending a $X^\mu\to E^{n+r}$ to an $X_+\to E^n$ on the level of spectra, too?

I am not asking for a proof of the Thom isomorphism. Even with the existence of such a map one has to prove that the correspondence induces an isomorphism on homotopy groups.