  We give a diagrammatic presentation in terms of generators mod relations of
the representation category of $U_q(\mathfrak{sl}_n)$. More precisely, we
produce all the relations among $\rm{SL}_n$-webs, thus describing the full
subcategory tensor-generated by fundamental representations $\bigwedge^k
\mathbb{C}^n$ (this subcategory can be idempotent completed to recover the
entire representation category). Our result answers a question posed by
Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim
[arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an
application of quantum skew Howe duality.
