Given a monoidal category $\mathscr{ C}$ with an object $J$, we construct a monoidal category $\mathscr{ C}[J^{ \vee}]$ by freely adjoining a right dual $J^{ \vee}$ to $J$. We show that the canonical strong monoidal functor $\Omega : \mathscr{ C}\to \mathscr{ C}[J^{ \vee}]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega : \mathscr{ C}\to \mathscr{ C}[J^{ \vee}]$ is fully faithful and provide coend formulas for homs of the form $\mathscr{ C}[J^{ \vee}](U,\Omega A)$ and $\mathscr{ C}[J^{ \vee}](\Omega A,U)$ for $A\in \mathscr{ C}$ and $U\in \mathscr{ C}[J^{ \vee}]$. If $\mathbb{ N}$ denotes the free strict monoidal category on a single generating object $1$ then $\mathbb{ N}[1^{ \vee}]$ is the free monoidal category $\mathrm{ Dpr}$ containing a dual pair $- \dashv +$ of objects. As we have the monoidal pseudopushout $\mathscr{ C}[J^{ \vee}] \simeq \mathrm{ Dpr} +_{ \mathbb{ N}} \mathscr{ C}$, it is of interest to have an explicit model of $\mathrm{ Dpr}$: we provide both geometric and combinatorial models. We show that the (algebraist's) simplicial category $\Delta$ is a monoidal full subcategory of $\mathrm{ Dpr}$ and explain the relationship with the free 2-category $\mathrm{ Adj}$ containing an adjunction. We describe a generalization of $\mathrm{ Dpr}$ which includes, for example, a combinatorial model $\mathrm{ Dseq}$ for the free monoidal category containing a duality sequence $X_0\dashv X_1\dashv X_2 \dashv \dots$ of objects. Actually, $\mathrm{ Dpr}$ is a monoidal full subcategory of $\mathrm{ Dseq}$.
Categories: math.CT