If $A$ and $B$ are two $k$ algebra's with a $\mathbb{Z}_2$-grading then I know that a $\mathbb{Z}_2$-graded structure can be defined on their tensor product. One does this by altering the multiplication to
$ (a\otimes b)(c\otimes d) = (-1)^{\operatorname{deg} b \cdot \operatorname{deg} c}(ac\otimes bd)$
My question is if there is a universal property that captures this altered tensor product of algebra's?
The grading is not that important; let's discuss the universal property of the tensor product $A \otimes_k B$ of ordinary $k$-algebras first. If $A$ and $B$ are both commutative this is the coproduct; however this is not true in the noncommutative case, where the coproduct is the free product. (Note that this is not the universal property of the tensor product of modules.)
So what is the universal property of the tensor product of not-necessarily-commutative $k$-algebras? It is the "commutative coproduct"; the universal $k$-algebra admitting a homomorphism from both $A$ and $B$ whose images commute. Personally I find this a little unsatisfying but here it is. There is a different universal property in the Morita bicategory, but that one has the downside of only describing the tensor product up to Morita equivalence rather than up to isomorphism.
The graded tensor product has the same universal property except that the meaning of "images commute" is different, since it means graded-commuting.