Is it possible to define a Berkovich analytic space via its functor of points?

Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space $X$ over $k$ as a functor $$X:(k-\mathbf{Aff})^{op}\longrightarrow\mathbf{Sets}$$ from the category of $k$-affinoid spaces into the category of sets that fulfills the following two conditions:

(1) The functor $X$ is a sheaf in the $G$-topology on $(k-\mathbf{Aff})^{op}$.

(2) Locally in the $G$-topology, the functor $X$ is isomorphic to the functor of points associated to an affinoid algebra.

I realize that this is probably not quite the right definition, since it fails to take into account the net-structure of an affinoid atlas of a Berkovich analytic space. (see Berkovich, Étale Cohomology for Non-Archimedean Analytic Spaces, Section 1.2)

Is it possible to add an additional piece of data to this definition to get a definition that is equivalent to Berkovich's?

yes-- given any two definitions A and B, you can always "modify" A in such a way that you get B (first erase A, then insert B instead) ... . $\endgroup$ – Stefan Kohl Nov 15 '13 at 16:48