  Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we
describe a "light leaves" algorithm to construct a basis of morphisms between
arbitrary tensor products of fundamental representations for
$\mathfrak{so}_5\cong \mathfrak{sp}_4$ (and the associated quantum group). Our
argument has very little dependence on the base field. As a result, we prove
that when $[2]_q\ne 0$, the Karoubi envelope of the $C_2$ web category is
equivalent to the category of tilting modules for the divided powers quantum
group $\mathcal{U}_q^{\mathbb{Z}}(\mathfrak{sp}_4)$.
