In chapter $5$ of Morita's Geometry of Differential Forms, he defines a real vector bundle over a smooth manifold $M$ as a triple $(E,\pi,M)$, where $E$ is another smooth manifold, $\pi \colon E \to M$ is smooth, each fiber $E_p = \pi^{-1}[p]$ is a real vector space, and there's the trivialization condition (which we won't need here I guess).
Then he procceds to do a lot of cool constructions such as the quotient bundle and the dual bundle. I understand everything except why
$$E/F = \bigsqcup_{p \in M} E_p/F_p \quad\mbox{and}\quad E^\ast =\bigsqcup_{p \in M}E_p^\ast$$ are smooth manifolds. Here, of course, $(E,\pi,M)$ is a vector bundle over $M$ and $(F,\pi,M)$ is a subbundle.
I think I must be missing some basic fact about smooth manifolds, but anyway, I need some push here.