Let $f:M\to N$ be a smooth map between smooth manifolds.
Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$.
where $\mathcal L(V, W)$ denotes the space of all the linear maps between vector spaces $V$ and $W$.
I ask this because I want to make the following statement:
Let $f:M\to N$ be a smooth map. Then the map $p\mapsto df_p$ is smooth.
($df_p$ lives in $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$).