  An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its
left dual and right dual objects are isomorphic. Given such an object and a
choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects
which intertwine with $P$ and $Q$ in a compatible manner. We show that this
category lifts the monoidal structure of $\mathcal{C}$ and the closed structure
of $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitable
colimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct a
family of Hopf monads on arbitrary closed monoidal categories $\mathcal{C}$. We
also introduce the pivotal cover of a monoidal category and extend our work to
arbitrary pivotal diagrams.
