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:



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> 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
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> $\iota(p,v) = (p,\lambda\_p(v)) $.
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> Prove that the map $(p\mapsto \lambda\_p): X \to \text{Hom}(V,W)$ is smooth.
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> 


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!



Smoothness of a family of maps induced from isomorphism of trivial bundles
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.


