This may be trivial question, but I am trying to clarify my understanding of this concept.
In the universal property of tensor product of $k$-vector spaces $V$ and $W$, it is stated that
it is a vector space $L$ with a map $\eta:V\times W\rightarrow L$ such that: given any vector space $U$ and a bilinear map $\phi$ from $V\times W$ into $U$, there is a unique linear map $\phi_1$ from $L$ to $U$ such that $\phi_1\circ \eta=\phi$.
Q. I did not see clearly, what $V\times W$ is treated? Is it considered as a set? Or as a vector space? What should we think of $V\times W$ in the definition of universal property? According to this, the map $\eta$ is set map or linear map from $V\times W$ to $L$ (tensor product)?
It's considered to be a set, named only to define what a bilinear map is as a map of sets. You may find this conceptually unsatisfying, because we seem to have left the category $\text{Vect}$. But this is necessary, because $\text{Vect}$ as a bare category only knows what a linear map is and to define tensor products we need to know what a bilinear map is, so we go back to $\text{Set}$ to talk about those.
(One way to fix this is to think of $\text{Vect}$ not as a category but as a multicategory, with multimorphisms given by the multilinear maps.)