Family of vector spaces I am interested in the following expression
"Let $W_i (i\in I)$ be a family of vector spaces..."
I came to understand, that this means the same thing as "that there is a function from an index set $I$
$$
W:I\to \mathbf{Vect}
$$
which "picks out" the required vector spaces." (in other words $W_i(i\in I)$ tells me to view the entire function $W:I \to \mathbf{Vect}$, not only its image.)
What I am unsure about is what the codomain space is. What is the space of all vector spaces? Or should I view the above $W$ as a functor rather than a function?
 A: You could view it as a functor from the set $I$ viewed as a discrete category into the category of vector spaces. This neatly hides away some set-theoretic concerns.
Alternatively, in a typical set-theoretic context, a function is (or can be) identified with its graph, so the function representing the family $W$ is just a set of $I$-many pairs whose second components are vector spaces. An individual vector space is (usually) a set by definition, so this is completely fine set-theoretically. 
Since the class of all vector spaces is a proper class, you can't (in ZFC) say $W\in (I\to\mathbf{Vect})$, but you can say $\mathsf{dom}(W)=I \land \forall i\in I. \mathsf{isVectSpace}(W(i))$ where $\mathsf{isVectSpace}$ is the predicate representing the class $\mathbf{Vect}$. Abstracting from $W$ (and also $I$ if you want), this gives you a predicate specifying that $W$ is a $I$-indexed family of vector spaces. This predicate represents the proper* class of $I$-indexed families of vector spaces. (Many people notationally overload $\in$ to also be used for membership in a class. Alternatively, in many set theories other than ZFC $\in$ does operate on classes or their analogues, but $I\to\mathbf{Vect}$ wouldn't mean anything. And then there are set theories (often used for the definition of categories) where there'd be almost no issue at all: you'd have a large set of vector spaces with small sets as carriers and the set of functions from a small set to a large set is a perfectly well-defined large set.)
To be clear, there are no set-theoretic concerns with the family $W$, only with the "collection" of all $I$-indexed families of vector spaces, i.e. the "collection" $W$ and things like it would be a "member" of.
* Assuming $I\neq\emptyset$, otherwise it's just the singleton set containing the empty function.
A: You have a set $I$, and for each $i \in I$ you have a vector space. Don't overthink it. While category theory has its place, there is very little motivation for thinking of every statement like this in terms of functors and categories.
A: Don't look too far. A family $\bigl(W_i\bigr)_{i\in I}$ of vector spaces is created as soon as  each teaching assistant $i\in I$ has selected a vector space $W_i$ in some way. You don't have to think of the universe of all vector spaces and an actual map, or production scheme, $I\to{\bf vect}, \ i\mapsto W_i$ here. This "map"  is pretty arbitrary, maybe "nonconstructible", and nobody asks about continuity and such. It only serves to organize the collection $\bigl(W_i\bigr)_{i\in I}\>$.
