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I'm reviewing a theorem from Golubitsky's Stable Mappings and Their Singularities, where it is proved that we can construct new vector bundles from old ones via smooth covariant functors. Anyway, as part of the proof there is this simple lemma:

Let $X$ be a smooth manifold with two trivial bundles $X \times V$ and $X\times W$ (with $V,W$ vector spaces) and let $\iota: X\times V \to X \times W$ be a bundle isomorphism. Then we may define a family of linear maps $\{ \lambda_p:V \to W\}_{p\in X}$ via the identification

$\iota(p,v) = (p,\lambda_p(v)) $.

Prove that the map $(p\mapsto \lambda_p): X \to \text{Hom}(V,W)$ is smooth.

This seems somehow clear-- the functions $\lambda_p$ must vary smoothly with $p$ since $\iota$ is smooth in both arguments--but the details are kinda eluding me.

For instance, it seems fairly clear that if $v$ is fixed then the map $(p \mapsto \lambda_p(v)): X \to W$ is smooth, but the present problem seems subtly different. Clearly I am confused about the details for which we 'build up' from smoothness of $\lambda_p(v)$ for each $v$ to smoothness of the present map. It's been a while since I've worked seriously with vector bundles-- anyway, if anyone can kindly help me to spell out these details, I'd greatly appreciate it!

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I assume that $V,W$ are finite-dimensional. The claim holds for any smooth bundle morphism $\varphi: X \times V \to X \times W$. Let $v_1,\dotsc,v_n$ be a basis of $V$. Then $K^n \to V$, $e_i \mapsto v_i$ is a linear diffeomorphism, which induces a linear diffeomorphism $\mathrm{Hom}(V,W) \cong W^n$. Thus, it suffices to prove that for every index $i$ the map $X \to W$ defined by the composition $$X \xrightarrow{~(\mathrm{id }_X,v_i)~} X \times V \xrightarrow{~\varphi~} X \times W \xrightarrow{~\mathrm{pr}_W~}W$$ is smooth, which follows since all these maps are smooth.

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