I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties.

In the literature (e.g. in Coleman-Mazur) I've seen people talk about the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$ (here $\mathbf{C}_p$ is the completion of an algebraic closure of $\mathbf{Q}_p$). Based on the case of schemes, if $X$ and $T$ are rigid spaces over $\mathbf{Q}_p$, I assume the set of $T$-valued points of $X$ is the set $\mathrm{Hom}_{\mathbf{Q}_p}(T,X)$, where the morphisms are in the category of $G$-topologized spaces over $\mathrm{Sp}(\mathbf{Q}_p)$. But $\mathbf{C}_p$ is not a $\mathbf{Q}_p$-affinoid algebra, so, strictly speaking, it's associated $G$-topologized space $\mathrm{Sp}(\mathbf{C}_p)$ is not a rigid space over $\mathbf{Q}_p$, right? I guess there is a morphism of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow\mathrm{Sp}(\mathbf{Q}_p)$, and so one could define the $\mathbf{C}_p$-points of $X$ in the same way as I did above, just morphisms of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow X$ compatible with the morphisms to $\mathrm{Sp}(\mathbf{Q}_p)$.

Question: Is this what is meant by the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?

The only alternative I can think of is to define the $\mathbf{C}_p$-points of a rigid space $X$ over $\mathbf{Q}_p$ as the $\mathbf{C}_p$-valued points of the base change $X_{\mathbf{C}_p}$. Although, if my intuition drawn from schemes is to be trusted, this is probably equivalent to the definition I suggested above.