A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $S$. We say a sheaf $X\colon S^{\text{op}}\to\mathsf{Set}$ is constant if it is the sheafification of a constant presheaf, i.e. one that factors through the terminal map $S^{\text{op}}\to \{*\}$.

But what if we forget the site $S$ and consider $X$ as an object in the topos? Can we characterize the property of $X\in\mathcal{E}$ being constant? More specifically, two questions:

1. Is the property of $X$ being constant dependent on the choice of site $S$?
2. Is there a way to characterize constant objects in the internal language of $\mathcal{E}$?