Let $(\mathcal{C},\mathcal{T})$ be a site and $\mathcal{X}$ a stack over it. Suppose $(\mathcal{C},\mathcal{T})$ subcanonical and hence for every object $U$ in $\mathcal{C}$ consider the associated stack $\overline{U}$ (here with stack I mean a category $\mathcal{X}$ fibred in groupoids over $\mathcal{C}$ with projection $p_{\mathcal{X}}:\mathcal{X}\to\mathcal{C}$ satisfying "sheaf-theoretic" gluing conditions). Objects in $\overline{U}$ are pairs $(X,\phi)$ where $X\in\mathcal{C}$ and $\phi:X\to U$. A morphism $(X_1,\phi_1)\to(X_2,\phi_2)$ is just a morphism $X_1\to X_2$ in $\mathcal{C}$ commuting with the $\phi_i$'s.
We say that a stack $\mathcal{X}$ is representable if it is equivalent to $\overline{U}$ for some $U\in\mathcal{C}$.
I was wondering if the following statement is right or not: given a stack $\mathcal{X}$ and objects $U,V\in\mathcal{C}$ with morphisms $F:\overline{U}\to\mathcal{X}$ and $G:\overline{V}\to\mathcal{X}$, then there exists an object $T\in\mathcal{C}$ such that the fibre product $\overline{U}\times_{F,\mathcal{X},G}\overline{V}$ is equivalent to $\overline{T}$.
In the realm of $C^{\infty}$-algebraic geometry (see http://arxiv.org/pdf/1001.0023v4.pdf ) I have met the following assertion: Let $\overline{\underline{*}}$ be the $C^{\infty}$-stack associated to the $C^{\infty}$-scheme $\rm{Spec}\,\mathbb{R}$ and $x:\overline{\underline{*}}\to\mathcal{X}$ a 1-morphism, then there exists a $C^{\infty}$-scheme $G$ such that $\overline{\underline{*}}\times_{x,\mathcal{X},x}\overline{\underline{*}}\simeq\overline{G}$ (Definition 8.20 in the link above). I have tried to see that this is true by a direct computation, taking as $G$ the $C^{\infty}$-scheme $p_{\mathcal{X}}(x(\underline{*},id_{\underline{*}})$, but I was not able to conclude. I am quite convinced that this is not the right choice. At the same time, my feeling is that this result is valid in general, but since I cannot solve even the particular case above, I am not sure.
Do you maybe can suggest a right choice of $G$ in the particular case or an alternative argument used to face this kind of problems? And also, is the first statement right in general?
Thank you very much!