Galactic nuclei typically host either a Nuclear Star Cluster (NSC, prevalent in galaxies with masses $lesssim 10^{10}M_odot$) or a Massive Black Hole (MBH, common in galaxies with masses $gtrsim 10^{12}M_odot$). In the intermediate mass range, some nuclei host both a NSC and a MBH. In this paper, we explore scaling relations between NSC mass (${cal M}_{rm NSC}$) and host galaxy total stellar mass (${cal M}_{star,rm gal}$) using a large sample of NSCs in late- and early-type galaxies, including a number of NSCs harboring a MBH. Such scaling relations reflect the underlying physical mechanisms driving the formation and (co)evolution of these central massive objects. We find $sim!1.5sigma$ significant differences between NSCs in late- and early-type galaxies in the slopes and offsets of the relations $r_{rm eff,NSC}$--${cal M}_{rm NSC}$, $r_{rm eff, NSC}$--${cal M}_{star,rm gal}$ and ${cal M}_{rm NSC}$--${cal M}_{star,rm gal}$, in the sense that $i)$ NSCs in late-types are more compact at fixed ${cal M}_{rm NSC}$ and ${cal M}_{star,rm gal}$; and $ii)$ the ${cal M}_{rm NSC}$--${cal M}_{star,rm gal}$ relation is shallower for NSCs in late-types than in early-types, similar to the ${cal M}_{rm BH}$--${cal M}_{star,rm bulge}$ relation. We discuss these results in the context of the (possibly ongoing) evolution of NSCs, depending on host galaxy type. For NSCs with a MBH, we illustrate the possible influence of a MBH on its host NSC, by considering the ratio between the radius of the MBH sphere of influence and $r_{rm eff, NSC}$. NSCs harbouring a sufficiently massive black hole are likely to exhibit surface brightness profile deviating from a typical King profile.
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