Let $M$ be a smooth manifold, let $(U_i)_{i\in I}$ be an open cover, let $F$ be a smooth manifold, and let $(g_{ij})$ be a smooth $\operatorname{Diff}(F)$-valued cocycle on this cover. Form the disjoint union

\begin{align*} \bigsqcup_{i\in I} U_i\times F \end{align*}

and impose the relation

\begin{align*} (i,x,y)\sim (j,x,g_{ji}(x)(y)) \end{align*}

whenever $x\in U_i\cap U_j$. If the resulting quotient is Hausdorff and second-countable, then it has a unique smooth fibre bundle structure over $M$ for which the canonical quotient maps $U_i\times F\to E$ induce local trivializations with typical fibre $F$ and transition functions $g_{ij}$. For a smooth $GL(k,\mathbb R)$-valued cocycle, the same construction gives a smooth rank $k$ vector bundle.