Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)_{top}$ the set of all equivalent Principal bundles on $M$ with structural group $G$ in the topological category when $G$ is seen as a topological group. Then we have a natural map $\varphi:Bun(M,G) \rightarrow Bun(M,G)_{top}$. Is $\varphi$ a bijection and how to prove it?