If $\mathcal{C}$ is a monoidal, $R$-linear category, what does the notation $\mathcal{C}\otimes_R\bar{R}$ mean, if $\bar{R}$ a quotient of $R$?

Question

Suppose $\mathcal{C}$ is a strict, monoidal $R$-linear category, where $R$ is some commutative ring. If $\bar{R}$ is a quotient ring of $R$, what does the notation $\mathcal{C}\otimes_R\bar{R}$ mean? Is it the category with the same objects, but the hom-spaces are now $\bar{R}$-modules, not $R$-modules?

Answer 1

Yes, it means the hom spaces have all been tensored with $\overline{R}$. (Note that this notation is consistent with the case that $C$ has one object, in which case it is a commutative ring.)