Let $X$ be a fixed parametrizing space. Let $E$ and $E'$ be two spectra and let $E_X$ and $E'_X$ be their trivial parametrized versions. Intuitively I imagine that the morphisms of parametrized spectra $E_X\to E'_X$ should correspond to maps $X\to \operatorname{Map}(E, E')$ where the mapping spac eon the right should be the underlining infinite loop space of the function spectrum, if you wish $\operatorname{Map}(E, E')=\Omega^\infty F(E, E').$

Is this, or at least anything similar to it, actually true?

Let $X$ be a fixed parametrizing space. Let $E$ and $E'$ be two spectra and let $E_X$ and $E'_X$ be their trivial parametrized versions. Intuitively I imagine that the morphisms of parametrized spectra $E_X\to E'_X$ should correspond to maps $X\to \operatorname{Map}(E, E')$ where the mapping space on the right should be the underlining infinite loop space of the function spectrum, if you wish $\operatorname{Map}(E, E')=\Omega^\infty F(E, E').$

Is this, or at least anything similar to it, actually true?