Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also this mathoverflow question by Martin Brandenburg.

My question is: can we consider a multiplicative system consists of both objects and morphisms in $\mathcal{C}$? This means that we have a collection of objects $x_i$ and a collection of morphisms $f_i$ such that $x_i \otimes x_j$ is still in the collection $x_i$ and $x_i$ and $f_j$ satisfies some "compatible condition". And can we define a localization along this more general multiplicative system?

Notice that in this viewpoint the case in the first paragraph can be considered as the multplicative system with only one object $1$ (and a system of morphisms).