The braid category is the free strict braided monoidal category on one generator

Define $F_X(n):=X^{\otimes n}$ on objects, with $F_X(0):=\mathbf{1}$. For each generator ${\sigma }_i\in B_n$, define $F_X({\sigma }_i)$ to be the morphism $1_X^{\otimes (i-1)}\otimes c_{X,X}\otimes 1_X^{\otimes (n-i-1)}:X^{\otimes n}\to X^{\otimes n},$ where the braiding acts only on the $i$th and $(i+1)$st tensor factors.

By [L3], the neighboring generators from step 1.1 satisfy the braid relation. Local braidings on disjoint tensor factors commute because in a strict monoidal category they act on separate coordinates. Therefore the Artin relations of $B_n$ hold, and [L4] extends the assignment of step 1.1 uniquely to a homomorphism $B_n\to \mathrm{Aut}(X^{\otimes n})$ for every $n$.

The family from step 2.1 defines a functor $\mathcal{B}\to \mathcal{C}$ because morphisms exist only between equal objects in $\mathcal{B}$, and group multiplication in each $B_n$ is respected by the constructed homomorphism. By construction $F_X(m+n)=F_X(m)\otimes F_X(n)$ on objects, the tensor of braids is sent to juxtaposition of local braidings, and the standard braiding ${\beta }_{m,n}$ of $\mathcal{B}$ is sent to the corresponding block braiding in $\mathcal{C}$. Hence $F_X$ is strict braided monoidal.

Any strict braided monoidal functor $\mathcal{B}\to \mathcal{C}$ sending $1$ to $X$ must send $n=1^{\otimes n}$ to $X^{\otimes n}$ and each generator ${\sigma }_i$ to the local braiding on the $i$th and $(i+1)$st factors. Since the ${\sigma }_i$ generate every $B_n$, such a functor must equal $F_X$. Therefore $F_X$ is unique.