$Z$ definition? (Hussemoller, fibre bundles)

Hussemoller doesn't describe what $$Z$$ is in this definition (page 67, third edition):

Note that $$\mathbf L(\mathbb F^m, \mathbb F^n)$$ is the collection of all linear functions $$\mathbb F^m \xrightarrow{\quad} \mathbb F^n$$. Also, have $$\mathbf{VB}_B$$ be the category of vector bundles over a space $$B$$. Finally, have $$\mathbf{VB}_0 (p, q)$$ be the product category of $$p$$ copies of $$\mathbf{VB}_B$$ and $$q$$ copies of the respective dual category $$\mathbf{VB}_B^*$$. Hussemoller does not give a clear definition of $$Z$$, and uses it as a placeholder for several different objects within the book. The most recent definition I could find is:

There is no clear way to reconcile this definition, since he nondescriptly mentions $$z$$ as it it were an element of $$Z$$, without describing the structure or what the map is computationally.

The solution is simple: $$Z$$ is any space. Husemoller should perhaps better have written
A functor $$F: \mathbf{VB}_0(p, q) \to \mathbf{VB}_0$$ is called continuous provided for each space $$Z$$ and each family of maps $$u_i : Z \to \mathbf{L}(V_i, W_i)$$, where $$1 \le i \le p + q$$, the function $$z \mapsto F(u_1(z), \ldots, u_{p+q}(z))$$ is a map $$Z \to \mathbf{L}(F(V_1,\ldots, V_p, W_{p+1},\ldots, W_{p+q}), F(W_j, \ldots, W_p, V_{p+1}, \ldots, V_{p+q}))$$
In the proof presented in your question $$Z$$ is specified to be the sum space of the family $$\{V_i \times G\}$$.