In this article, there are two proposition as the following:
1.(Proposition.) For every bilinear map $f:V\times W\to U$ there is a unique linear map $h:V\otimes W\to U$ such that $hg=f$, where $g$ is the bilinear map from $V\times W$ to $V\otimes W$ that takes $(v,w)$ to $v\otimes w$.
2.(Lemma.) 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 $U\otimes V$.
The author says in the remark that "the point of the lemma is that any bilinear map $b:U\times V\to X$ satisfying the universal property is isomorphic to the map $g:U\times V\to U\otimes V$ in an obvious sense."
As I understand, when people talk about isomorphism, there should be two structures. Here comes my question:
What does "isomorphic" mean in this context? And where are the underlying two structures? (How can a map isomorphic to another map?)