In this answer by Zhen Lin it is stated the left $G$-set $G$ (where $G$ acts on the left by translation) is unique among all left $G$-sets as the only object $X$, up to isomorphism, in the category $G\mathbf{Set}$ such that the corepresented functor $$G\mathbf{Set}(X,-):G\mathbf{Set}\longrightarrow\mathbf{Set}$$ commutes to colimits. Upon solving this exercise, one sees furthermore that the objects that commute to coproducts are the left $G$-sets containing a single orbit.
Does this property ($X\in\mathrm{ob}(C)$ such that $C(X,-)$ commutes to colimits) have a name ? Are there other enlightening instances of this property that connect it to other notions, maybe in general presheaf categories, or in spaces or module categories ? For instance, this notion seems to be related to that of compact objects.
Such objects are called tiny. A useful generalization of the fact you quote is that the tiny objects in the presheaf category $[C^{op}, \text{Set}]$ are precisely the retracts of the representable presheaves. Also, in a cocomplete abelian category this condition is equivalent to being compact projective. See this blog post for more.
The condition that $\text{Hom}(X, -)$ commutes with coproducts is called connected.