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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 $\U(\fg)$.

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