On $g$-finiteness in the category of projective presentations

We provide new equivalent conditions for an algebra $\mathrm{\Lambda }$ to be $g$-finite, analogous to those established by L. Demonet, O. Iyama, and G. Jasso, but within the category of projective presentations ${\mathcal{K}}^{[-1,0]}(\text{proj}\mathrm{\Lambda })$. We show that an algebra has finitely many isomorphism classes of basic $2$-term silting objects if and only if all cotorsion pairs in ${\mathcal{K}}^{[-1,0]}(\text{proj}\mathrm{\Lambda })$ are complete. Furthermore, we establish that this criterion is also equivalent to all thick subcategories in ${\mathcal{K}}^{[-1,0]}(\text{proj}\mathrm{\Lambda })$ having enough injective and projective objects.