Consider a symmetric monoidal category $(C, \otimes, I)$, where $I$ is the unit. Then there is a restricted Yoneda functor $$ C \rightarrow Hom(I,I)-mod $$ taking an object $X$ of $C$ to $Hom(I, X)$, viewed as a right $Hom(I, I)$-module. I am wondering when this functor is full and faithful. This seems to be the case when $C$ is $R$-mod for a commutative ring $R$ and $\otimes=\otimes_R$, but maybe this case is special since $R$ is also a generator. There are some counterexamples pointed out here: https://mathoverflow.net/questions/47342/tensor-variant-of-mitchells-embedding-theorem/47368#47368 for example when $C$ is complex representations of a finite group with tensor product given by tensor product of representations. (also see Can tensor abelian categories always be embedded into the category of modules?)
Are there any mild conditions I can add to make this functor full and faithful? Really, I would like to work with stable infinity categories (or at least with triangulated categories over $\mathbb{Z}$), so maybe the claim has some hope in those settings? (at least, I believe it rules out Lurie's counterexamples).