Braidings of Tensor Spaces

Let $V$ be a braided vector space, that is, a vector space together with a solution $\hat{R}\in {\text{End}}(V\otimes V)$ of the Yang--Baxter equation. Denote $T(V):=\bigoplus_k V^{\otimes k}$. We associate to $\hat{R}$ a solution $T(\hat{R})\in {\text{End}}(T(V)\otimes T(V))$ of the Yang--Baxter equation on the tensor space $T(V)$. The correspondence $\hat{R}\rightsquigarrow T(\hat{R})$ is functorial with respect to $V$.