We will use the following terminology:
A morphism \(F\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) is called faithful if its underlying \((\infty,2)\)-functor is (see notation 5.3.3). It is called surjective-on-objects-and-dominant-on-1-morphisms if it is surjective on objects and if for each \(c, c'\in \mathcal C\), the induced additive functor \(\underline{\mathrm{Hom}}_{\mathcal C}(c,c') \rightarrow\underline{\mathrm{Hom}}_{\mathcal C}(Fc,Fc')\) is dominant.
A morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\) is called faithful or surjective-on-objects-and-dominant-on-1-morphisms if the underlying morphism in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) is.