$\DeclareMathOperator{\Sp}{\mathrm{Sp}}$I am taking a special case $\Sp$ here, mainly because it has nice categorical properties.

Let $R$ be an $E_\infty$-ring spectrum. In Higher Algebra, Lurie proves we have a forgetful functor (part of monadic adjunction) $$ U_R:\operatorname{Mod}_R(\Sp) \rightarrow \Sp$$ where $\Sp$ is in the $\infty$-category of spectra.

$U_R$ reflects equivalences. But is $U_R$ faithful in the sense that that the induced map of $$Map(x,y)\rightarrow Map(U_Rx,U_ry)$$ mapping spaces is $0$-truncated in the $\infty$-category of spaces. i.e. the homotopy fibers are $0$-truncated.

One categorically, $U$ is faithful in many cases, i.e. if we replace $\Sp$ with $\mathrm{Ab}$. Perhaps the answer is false in $\infty$-categories. I'd like to understand what goes wrong. Some comments on the following would be helpful:

• A counter example where $U_R$ is not faithful. (i.e. is it faithful when $R=H\Bbb Z$? )
• A brief/reference explanation for what accounts of this.