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We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category 
$\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category 
$\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ of quantum 
$\mathfrak {sp}_{6}$ representations generated by the fundamental representations, for generic values of the parameter q. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor 
$\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$, that all 
$\textrm {Hom}$-spaces in 
$\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in 
$\textbf {Web}(\mathfrak {sp}_{6})$ is one-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum 
$\mathfrak {sp}_{6}$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.
