Abstract The first word in the title is intended in a sense suggested by Lawvere and Schanuel whereby finite sets are objective natural numbers. At the objective level, the axioms defining abstract Mackey and Tambara functors are categorically familiar. The first step was taken by Harald Lindner in 1976 when he recognized that Mackey functors, defined as pairs of functors, were equivalently single functors with domain a category of spans. In 1993, Tambara recognized that TNR-functors (that is, functors designed to have abstract trace, norm and restriction operations, and now called Tambara functors) were equivalently certain functors out of a category of polynomials. We define objective Mackey and objective Tambara functors as parametrized categories that have local finite products and satisfy some parametrized completeness and cocompleteness restriction. However, we can replace the original parametrizing base for objective Mackey functors by a bicategory of spans while the replacement for objective Tambara functors is a bicategory obtained by iterating the span construction; these iterated spans are polynomials. There is an objective Mackey functor of ordinary Mackey functors. We show that there is a distributive law relating objective Mackey functors to objective Tambara functors analogous to the distributive law relating abelian groups to commutative rings. We remark on hom enrichment matters involving the 2-category $\textrm{Cat}_{+}$ of categories admitting finite coproducts and functors preserving them, both as a closed base and as a skew-closed base.

Abstract 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.