Pairings and unital multiplication of sequential prespectra

Let $E,F,G$ be sequential prespectra, with structure maps denoted ${\sigma }^E,{\sigma }^F,{\sigma }^G$. A pairing $E\wedge F\to G$ is a family of based maps

${\mu }_{m,n}:E_m\wedge F_n⟶G_{m+n}$

together with based homotopies making it compatible with each structure coordinate. Suppressing only the canonical associators, the two required comparisons are

${\mu }_{m+1,n}({\sigma }_m^E\wedge 1)\simeq {\sigma }_{m+n}^G(1_{S^1}\wedge {\mu }_{m,n})$

on $S^1\wedge E_m\wedge F_n$, and

${\mu }_{m,n+1}(1\wedge {\sigma }_n^F)\simeq {\sigma }_{m+n}^G(1_{S^1}\wedge {\mu }_{m,n})({\tau }_{E_m,S^1}\wedge 1)$

on $E_m\wedge S^1\wedge F_n$. Here ${\tau }_{E_m,S^1}$ moves the new suspension coordinate past $E_m$; this twist is part of the formula, not an implicit sign.

A unital multiplication on $E$ is such a pairing $E\wedge E\to E$, a based unit $\eta :S^0\to E_0$, and specified compatible homotopies from the two unit composites to the identity and between $\mu (\mu \wedge 1)$ and $\mu (1\wedge \mu )$, with the canonical associators inserted. A commutativity datum additionally gives compatible comparison homotopies between ${\mu }_{m,n}$ and ${\mu }_{n,m}{\tau }_{E_m,E_n}$, including the permutation of the $m$ and $n$ suspension-coordinate blocks. That block permutation has degree $(-1{)}^{mn}$ on $S^{m+n}$. All homotopies are required to respect the two displayed structure comparisons.