Let $q: E \to Y$ be a smooth real vector bundle over a smooth manifold $Y$. Let $X$ be a paracompact smooth manifold, let $I = [0,1]$, and let $F: X \times I \to Y$ be a smooth map. For $i \in \{0,1\}$, define the endpoint maps $f_i: X \to Y$ by $f_i(x) = F(x,i)$. Then the pullback vector bundles $f_0^*E \to X$ and $f_1^*E \to X$ are isomorphic as smooth vector bundles over $X$.