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Let $\omega$ be the universal connection on the bundle $EU\left(n\right)\to BU\left(n\right)$. Given a principal $U\left(n\right)$-bundle $P\to M$ with connection $A$, we determine the homotopy type of the space of maps $\varphi$ of $M$ into $BU\left(n\right)$ such that $\left({\varphi }^{+}EU\right(n),{\varphi }^{+}\omega )$ is isomorphic to $(P,A)$. Here ${\varphi }^{+}$ denotes pull-back.