The tensor product as a universal morphism Given two vector spaces $V,W$ (say, over $\mathbb{C}$), we can form their tensor product $V \otimes W$, and we get a (bilinear) map $i : V \times W \rightarrow V \otimes W : (v,w) \mapsto v \otimes w$. This construction is characterized by the following universal property.

For vector space $A$ and any bilinear map $\phi : V \times W \rightarrow A$, there exists a unique linear map $T : V \otimes W \rightarrow A$ such that $\phi = T \circ i$.


I'm starting to learn about universal morphisms in category theory, and I thought this would be a good example to try to work through the definition. In particular, I would think that $(V \otimes W, i)$ should be a universal morphism from $V \times W$ to a certain functor $\Gamma$, as illustrated by the following diagram.

Looking at it closely, we need to consider a category $\mathcal{C}$ and a functor $\Gamma : \textbf{Vect} \rightarrow \mathcal{C}$. We should be able to think of $V \times W$ as an object of $\mathcal{C}$, and we should be able to think of the arrows of $\mathcal{C}$ as bilinear maps. Again, the goal is for $(V \otimes W, i)$ to be a universal morphism from $V \times W$ to $\Gamma$. Then the tensor product would be the left adjoint of $\Gamma$.
However, I can't seem to figure out what $\mathcal{C}$ and $\Gamma$ should be. My original idea was that the objects of $\mathcal{C}$ are pairs of vector spaces, and arrows $(V,W) \rightarrow (X,Y)$ in $\mathcal{C}$ are bilinear maps $V \times W \rightarrow X \times Y$. But, then I can't find a suitable functor $\Gamma$.
Can we define $\mathcal{C}$ and $\Gamma$ that fulfill these conditions?
 A: $\newcommand{\v}{\mathsf{Vect}}$Fix a field $k$. Given a pairing of vector spaces $(V,W)$, the bilinear map $(V\times W\to V\otimes W)$ is an initial (universal) object of the category with objects vector spaces $U$ and bilinear maps $V\times W\to U$.
If you want a universal morphism, we can do this via an adjunction. This is highly artificial, and in my view the category I’m about to outline is not noteworthy, but it suffices to describe what you want in terms of universal morphisms of functors. As commented, universal properties aren’t always usefully described in this way - the only case of “universal morphism” that appears in my experience is the case of universal morphisms pertaining to adjunctions.
Let’s define $\v_k$ to be the usual category of vector spaces over $k$, and define $\v_k^\ast$ to be the same category augmented with bilinear maps.
Specifically, let $\v_k^\ast$ have objects all $k$-vector spaces as well as formal pairs $(V,W)$ of spaces. The arrows between spaces $A\to B$ are just the linear maps; the arrow class $A\to(V,W)$ is empty; the arrow class $(V,W)\to A$ is the set of bilinear maps $V\times W\to A$, and the arrow class $(V,W)\to(X,Y)$ contains formal pairs $(f,g)$ of linear maps $V\to X,W\to Y$. All compositions are in the obvious way.
Let $\Gamma:\v_k\hookrightarrow\v_k^\ast$ be the obvious inclusion of categories. We want to define a reflection to this, a left adjoint $\Lambda:\v_k^\ast\to\v_k$. By the beautiful machinery of adjunctions, it is sufficient to specify initial, universal morphisms. If $A\in\v_k^\ast$ is a single vector space, let $\Lambda A=A$ and $\eta_A:A\to\Gamma\Lambda A=A$ is the identity arrow (in $\v_k^\ast$). If $(V,W)$ is a pair of vector spaces, let $\Lambda(V,W)$ be $V\otimes_k W$, and $\eta_{(V,W)}:(V,W)\to\Gamma\Lambda(V,W)=V\otimes W$ is the universal tensor bilinear map.
You can check the $\eta_\bullet$ are all initial morphisms. By magic, this is enough to assemble to a natural transformation, the unit, induce a dual counit transformation, and make $\Lambda$ functorial, so that $\Lambda\dashv\Gamma$.
