Classification theorem of topological vector bundles via presheaves

Question

My issue is mainly set theoretical:
In "Dale Husemoller: Fiber Bundles" it says on page 34 that there is a cofunctor $Vekt_{k}: P\rightarrow ens$, where $P$ is the category of paracompact spaces with homotopy classes of maps and ens the category of sets and functions, assigning to each paracompact space $B$ the set of $B$-isomorphism classes of vector bundles over $B$.
Unfortunately,I don't have any profound knowledge of set theory and I do not understand why this actually forms a set and not a proper class. Can anyone help me with this?

Answer 1

I assumed your vector bundles are finite-dimensional, because the isomorphism classes of infinite-dimensional vector bundles never form a set-not even over a point! Given that, the key point is that (real or complex) vector bundles over $B$ have bounded cardinality: their underlying sets cannot be larger than the maximum of the cardinality of $\mathbb{R}$ and the cardinality of the base space. There are only a set worth of isomorphism classes of triples $(E,T,f)$ where $E$ is a set of bounded cardinality, $T$ is a topology on $E$, and $f$ is a continuous function $E\to B$. Indeed, we could assume that $E$ is always a subset of some fixed set by applying an appropriate isomorphism. The set of isomorphism classes of vector bundles is a subset of this set.