Let $\mathbb{A}$ and $\mathbb{B}$ be logics.

A meta relation or horizontal profunctor is

a matrix category

$\mathcal{R}(\underline{\mathbb{A}}, \underline{\mathbb{B}}):\mathbb{M}\mathrm{at}\mathbb{C}\mathrm{at}$

with matrix functors

left/precompose action

$\circ_\mathbb{A}: \mathbb{A}\otimes \mathcal{R}\to \mathcal{R}$

right/postcompose action

$\circ_\mathbb{B}: \mathcal{R}\otimes \mathbb{B}\to \mathcal{R}$

with invertible matrix transformations

center associator

$\alpha_\mathcal{R}: \mathbb{A}\circ (\mathcal{R}\circ \mathbb{B})\cong (\mathbb{A}\circ \mathcal{R})\circ \mathbb{B}$

left associator

$\alpha_\mathbb{A} : (\mathbb{A}\circ \mathbb{A})\circ \mathcal{R} \cong \mathbb{A}\circ (\mathbb{A}\circ \mathcal{R})$

right associator

$\alpha_\mathbb{B} : \mathcal{R}\circ (\mathbb{B} \circ \mathbb{B})\cong (\mathcal{R}\circ \mathbb{B}) \circ \mathbb{B}$

left unitor

$\upsilon_\mathbb{A}: \mathcal{R}\cong \mathrm{id}.\underline{\mathbb{A}}\circ \mathcal{R}$

right unitor

$\upsilon_\mathbb{B}: \mathcal{R}\cong \mathrm{id}.\underline{\mathbb{B}}\circ \mathcal{R}$

satisfying associator coherence