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

But what if we forget the site $Q$ 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 $Q$?
 2. Is there a way to characterize constant objects in the internal
    language of $\mathcal{E}$?