Suppose I have manifolds $Y,Z$ and a bundle $E \to Z$. I want to consider pairs $(\phi,\psi)$, where
- $\phi: Y \to Z$ is a smooth map,
- $\psi: Y \to \phi^*E$ is a section of the pullback of $E$ under $\phi$.
Is the set of such pairs some sort of fiber product? It feels obvious that it is, but I seem to be missing it. In other words, I want a more intrinsic way of writing down the set $$ \{ (\phi,\psi) \mid \phi \in \mathsf{C}^\infty(Y;Z), \psi \in \Gamma(\phi^*E) \}. $$ Thanks for the help!