Let $B$ and $F'$ be smooth Hausdorff second-countable manifolds, let $G$ be a Lie group acting smoothly on $F'$, and let $(g_{ij})$ be a smooth $G$-valued cocycle on an open cover $(U_i)_{i \in I}$ of $B$. Then the associated fibre construction gives a smooth fibre bundle over $B$ with fibre $F'$ and structure group contained in the image of $G\to \operatorname{Diff}(F')$.