[001Y] Remark 1·i (M criterion).

If $\mathbb{K}=\Cat$, $\otimes : i_0\circ d_0\To i_1 \circ d_1$ amounts to the natural transformation

$$ \tilde\otimes : U^oV = d_0d_1^o \To i_0^oi_1 = ST^o $$

that is such that, for given $\alpha,\beta$,

$$ \tilde\alpha * \tilde\otimes* \tilde\vf = \tilde\alpha \qquad \tilde\vf * \tilde\otimes * \tilde\beta = \tilde\beta. $$

However, since $PQ^o$ is a generic profunctor, as well as $F^oG$, we can take $PQ^o=U^oV$, $\tilde\alpha = S$, $F^oG=ST^o$, $\tilde\beta = 1$, hence $\tilde\otimes = \tilde{\vf}^{-1}$, so $\vf$ is exact. We can observe directly, using 1·f [001V], that if $\vf$ is exact, we can define the multiplicative structure $\nu\otimes_\vf \mu := (\sum_{F,G}\bar\vf)^{-1}\nu * \bar\mu$. From this we deduce the following multiplicative criterion: a square $\vf : US \To VT$ in $\Cat$ is exact if and only if it admits a multiplicative structure $\otimes_\vf$ (so, $\otimes_\vf$ is unique, given by the inverse of $\tilde\vf$).