How should I understand "every bilinear function $f$ on $V\times W$"?

Question

In this article, there is a lemma as following:

Let $U$ and $V$ be vector spaces, and let $b:U\times V\to X$ be a bilinear map from $U\times V$ to a vector space $X$. Suppose that for every bilinear map $f$ defined on $U\times V$ there is a unique linear map $c$ defined on $X$ such that $f=cb$. Then there is an isomorphism $i:X\to U\otimes V$ such that $u\otimes v=ib(u,v)$ for every $(u,v)$ in $U\otimes V$.

There are several other places in this article where the author uses "every bilinear map $f$ defined on $U\times V$ " without specifying the range of the function.

As I understand, it may mean one of the following items:

1. $\forall f\in{\mathcal L}(V,W;Y)$ for some vector space Y;
2. $\forall f\in \bigcup_{Y\text{is a vector space}}{\mathcal L}(V,W;Y).$

Which one is correct and how should I understand it?

Answer 1

The second one. The statement is about the universal property and is essentially categorical. A simple reading of the relevant sentence should be that

$\forall Y$ a vector space and $\forall f:U\times V \to Y$ $\exists ! c: X\to Y$ s.t. $f = cb$.

Answer 2

The first statement would usually be expressed by something like "Suppose that there is a vector space $Y$ such that for every bilinear map $f$ from $U\times V$ to $Y$..." -- one wouldn't usually imply such a distinguished object without introducing it. In a sense, the target space $Y$ is a free variable of the statement, and free variables usually imply a universal quantifier applied to them (2), not an existential quantifier (1).