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Given a monoidal category 
$\mathcal C$ with an object J, we construct a monoidal category 
$\mathcal C[{J^ \vee }]$ by freely adjoining a right dual 
${J^ \vee }$ to J. We show that the canonical strong monoidal functor 
$\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ 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 
$\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ is fully faithful and provide coend formulas for homs of the form 
$\mathcal C[{J^ \vee }](U,\,\Omega A)$ and 
$\mathcal C[{J^ \vee }](\Omega A,U)$ for 
$A \in \mathcal C$ and 
$U \in \mathcal C[{J^ \vee }]$. If 
${\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then 
${\rm{N[}}{{\rm{1}}^ \vee }{\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout 
$\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X0 ˧ X1 ˧ X2 ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.