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!