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Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we can chose a connection $B$ on $P$ and define an isomorphism $\phi:f^{*}P\rightarrow g^{*}P$ by $\phi(x,p):=(x,\tilde H(1,x))$. Here, $\tilde H(t,x)$ is the lift of $t\rightarrow H(t,x)$, x fixed, induced by $B$.

My questions: is there a gauge transformation taking $\phi^{*}g^{*}B$ to $f^{*}B$?

Thank you very much.

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