TITLE: How to understand the Deligne' tensor product of finite abelian category
QUESTION [6 upvotes]: In the sec 1.11. "Delignes' tensor product of locally finite abelian categories" of the book "Tensor Categories" of EGNO, the deligne's tensor 
product $C \boxtimes D$ of two k-linear locally finite abelian categories $C$ ad $D$ is defined by the universal property:
$$
\matrix{
C \times D & \xrightarrow{\boxtimes} & C \boxtimes D\\
& F \searrow & \downarrow{ \exists ! G} \\
& & A
}
$$
i.e., $\boxtimes: C \times D \rightarrow C \boxtimes D$ is a bifuntor which is right exact in both variables and is such that
for any right exact bifunctor $F: C \times D \rightarrow A$, where $A$ is k-linear locally finite abelian category, there exists a unique right exact functor $G: C \boxtimes D \rightarrow A$ such that
$$
G \circ \boxtimes = F.
$$
Let $Vec$ be the category of finite dimensional $k$-vector spaces. By the proof provided in the book, $Vec \boxtimes Vec \cong Vec$ and $\boxtimes$ is give by the tensor product of vector spaces. To my understanding, if $B$ is a category which is 
equivalent to $C \boxtimes D$, then $B$ can also be viewed as the Delignes tensor product of $C$ and $D$.
Let $sVec$ be the skeleton of $Vec$. Then there should exists a functor 
$G: sVec \rightarrow Vec$ such that the following diagram commutes:
$$
\matrix{
Vec \times Vec & \xrightarrow{\otimes} & sVec \\
& \otimes \searrow & \downarrow{ G} \\
& & Vec
}.
$$
However, this is clearly impossible. So my question is if we can 
view $sVec$ as $Vec \boxtimes Vec$?

REPLY [7 votes]: The universal property isn't a characterization of $C \boxtimes D$, per se: the universal property is a property of the pair $(C \boxtimes D, \boxtimes)$ as an object of the coslice 2-category whose objects are pairs consisting of a category $B$ and a functor $C \times D \to B$. Now, $\mathsf{Vec}$ and $\mathsf{sVec}$ are equivalent just as categories, but there's no way to make them equivalent in this coslice 2-category (being equivalent here is a stronger condition). Does this help?