Let $\U_q(\fg)$ be the quantum supergroup of $\gl_{m|n}$ or the
modified quantum supergroup of $\osp_{m|2n}$ over the field of rational functions in $q$,
and let $V_q$ be the natural module for $\U_q(\fg)$.
There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to
the category of finite dimensional representations of $\U_q(\fg)$, which preserves ribbon category structures.
We show that
this functor is full in the cases $\fg=\gl_{m|n}$ or $\osp_{2\ell+1|2n}$.
For $\fg=\osp_{2\ell|2n}$,
we show that the space $\Hom_{\U_q(\fg)}(V_q^{\otimes r}, V_q^{\otimes s})$ is
spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for
two versions of the quantisation of $\U(\fg)$.