We will use the following terminology:
A functor \(F\colon \mathcal C\rightarrow\mathcal D\) between idempotent-complete \(\infty\)-categories is called dominant if every object in \(\mathcal D\) is a retract of an object in the image of \(F\).
A morphism in \(\mathrm{add}_{k}^{B\mathbb{Z}}\) is dominant, resp. fully faithful, if its underlying functor is.