We denote the bicategory of bimodules (or profunctors, or distributors) as $\Prof$, and its composition [of 1-cells, TN] as $\otimes$.

We know (cf (Bénabou, 1973) (Gouzou & Grunig, 1973) (Harting, 1977) (Thiébaud, 1971)) that the immersion $\Cat \to\Prof : F\mapsto \Cat[\firstblank, F\sndblank]$ is 2-full, and that $F\approx \Cat[\firstblank, F\sndblank] \dashv \Cat[F\firstblank, \sndblank] =: F^o$, in such a way that if $\vf : U\circ S \To V\circ T$ is a square in $\Cat$, it determines a 2-morphism in $\Prof$ $\tilde\vf : S\otimes T^o \To U^o\otimes V$, with components