talks are invariant) For any $s\in S$ and $e\in E$, if $s\cdot e$ is defined then $\pi(s\cdot e)=\pi(e)$.

4.  (universal on stalks) For any $s\in S$ and $x\in X$, $(E_x,\mu|_{S\times E_x})$ (which is well-defined by (U3)) is a universal $S$-set.

5.  (continuous) The partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous ($S$ is considered as a discrete space and $S\times E$ as a product space).

It is easy to see that (U4) implies (U1), so that (U1) may be omitted from the above list.

Let $\pi \colon E\to X$, $\pi'\colon E'\to X$ be étale spaces and $(E,\mu)$, $(s,e)\mapsto s\cdot x$, if defined, $(E',\nu)$, $(s,e)\mapsto s\circ x$, if defined, be universal $S$-sets in the topos ${\mathsf{Sh}}(X)$. A morphism $$f\colon (E,\mu) \to (E',\nu)$$ is defined as a morphism $f\colon E\to E'$ of étale spaces (that is, a continuous map such that $\pi=\pi'f$, cf. [@MM]) which is simultaneously a morphism of $S$-sets (that is, if $s\cdot e$ is defined then $s\circ f(e)$ is defined and $f(s\cdot x)=s\circ f(x)$). We denote the category of universal $S$-sets in the topos ${\mathsf{Sh}}(X)$ by ${\mathsf{Univ}}(S,X)$.

\[th:sheaves\] There is an equivalence of categories $${\mathsf{Prin}}(S, X) \,\,
{\mathrel{
  \settowidth{\@tempdima}{$\scriptstyle\tau$}
  \settowidth{\@tempdimb}{$\scriptstyle\rho$}
  \ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
  \mathop{\vcenter{
    \offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
    \rightarrowfill\cr\noalign{\kern.3ex}
    \leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}} \,\, {\mathsf{Univ}}(S,X).$$

We begin with the construction of the functor $$\tau\colon  {\mathsf{Prin}}(S, X)\to {\mathsf{Univ}}(S,X).$$ Let $E\colon L(S) \to {\mathsf{Sh}}(X)$ be a principal bundle over $X$ and $x\in X$. We first describe the colimit sheaf $\widetilde{E}\in {\mathsf{Sh}}(X)$. By definition, for each $x\in X$ we have a filtered functor $f_x\colon L(S)\to {\mathsf{Sets}}$ obtained by restricting $E$ to the stalks over $x$. We now apply the functor $\Psi$