  We define a $\mathbb{C}(q)$-linear pivotal category
$\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full
subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$
tensor-generated by the fundamental representations. This answers the type $C$
case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2
Lie algebras" (arXiv:q-alg/9712003).
