TITLE: Is the tensor product of non-commutative algebras a colimit?
QUESTION [5 upvotes]: For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of associative $R$-algebras, but it does satisfy a universal property, as given on Wikipedia. Is this some sort of colimit? If not, is there are a straightforward description of this universal property via a functor (right?) adjoint to the tensor product, as is the case for tensor products of modules?

REPLY [2 votes]: There is a tensor-hom adjunction for the tensor product of algebras, but it exists at the level of the Morita 2-category, rather than the 1-category of algebras. 
Namely, the Morita 2-category has objects $k$-algebras, and the category of morphisms $A \to B$ is the category $\text{Mod}(A^{op} \otimes B)$ of $(A, B)$-bimodules, where composition is given by tensor product. (All tensor products are over $k$.) The Morita 2-category has an internal hom $[A, B] = A^{op} \otimes B$, and its left adjoint is the tensor product in the sense that we have natural identifications
$$[A \otimes B, C] \cong A^{op} \otimes B^{op} \otimes C \cong [A, B^{op} \otimes C] \cong [A, [B, C]].$$
In the Morita 2-category the coproduct of two algebras $A$ and $B$ is $A \times B$ (rather than the free product), and tensor product distributes over this.