Let $(g_{ij})$ and $(g'_{ij})$ be smooth $G$-valued cocycles on the same open cover $(U_i)$ of a smooth base manifold $M$, where $G\subseteq \operatorname{Diff}(F)$ is a structure group. The corresponding reconstructed bundles are isomorphic by an isomorphism preserving the base and the structure group if there exist smooth maps $a_i:U_i\to G$ such that

\begin{align*} g'_{ij}(x)=a_i(x)\circ g_{ij}(x)\circ a_j(x)^{-1} \end{align*}

for every $x\in U_i\cap U_j$. For vector bundles, this formula is

\begin{align*} g'_{ij}(x)=a_i(x)g_{ij}(x)a_j(x)^{-1}. \end{align*}