The coproduct of all of the tensor powers of $V$ naturally inherits the structure of a monoid in $C$. This is called the tensor algebra of $V$. This is the free monoid object on $V$. For more on this see category of monoids.

Examples

Often in the literature this is considered for the case $C =$Vect of vector spaces. Given a vector space$V$, the $n$-fold tensor product of this space with itself, $V \otimes \cdots \otimes V$, is usually denoted $V^{\otimes n}$ and called the $n$th tensor power of $V$.