 to each $f_x$. Proposition \[prop:ff\] ensures us that each $\Psi(f_x)$ is a universal $S$-set. Note that each of the sets $\Psi(f_x)$ is non-empty by (PB1). As the stalks of the colimit sheaf are colimits of stalks, we may set $$\widetilde{E}=\bigcup_{x\in X} \Psi(f_x)$$ to be a disjoint union of all the sets $\Psi(f_x)$. The projection map $$p\colon \widetilde{E}\to X$$ given by $y\mapsto x$ if $y\in \Psi(f_x)$.

We may identify each space $E(e)$ with its image under the inclusion into $\widetilde{E}$. The colimit topology on $\widetilde{E}$ is the finest topology which makes all inclusion maps $E(e)\hookrightarrow \widetilde{E}$ continuous. The base of this topology is formed by the sets $A\subseteq \widetilde{E}$ such that $A\subseteq E(e)$ for some $e$. The structure of an $S$-set, $(s,y)\mapsto s*y$, where defined, on $\widetilde{E}$ is induced by the structures of $S$-sets on each on $\Psi(f_x)$, given by .

We now prove that for each $s\in S$ the set $$D_s=\{y\in \widetilde{E}\colon s*y \text{ is defined}\}$$ is open. Clearly, $D_s=D_{{\mathbf d}(s)}$. Let $y\in D_{{\mathbf d}(s)}$ and $A$ be a neighbourhood of $y$ in $\widetilde{E}$. Since the inclusion map $i\colon E_{{\mathbf{d}}(s)}\hookrightarrow \widetilde{E}$ is open, we have that $ii^{-1}(A)$ is a neighbourhood of $y$ in $\widetilde{E}$ which is contained in $D_{{\mathbf d}(s)}$. This implies that the set $D_{{\mathbf d}(s)}$ is open. Using the fact that the translation maps are continuous, it is routine to verify that the partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous.

We now define $\tau$ on morphisms. Let $E$ and $E'$ be principal $S$-bundles and $\varphi\colon E\to E'$ be a natural transformation. The family of continuous maps $E(e)\to E'(e)$ for each $e\in E(S)$ and the construction of $\widetilde{E}$ yield a continuous map $\tau(\varphi)\colon \widetilde{E}\to \widetilde{E'}$ which obviously satisfies the definition of a morphism of universal representations.

We now turn to the construction of the functor 