Let $\mathbb{X,Y,A,B}$ be logics.

Let $f: \mathbb{X}\,|\, \mathbb{A}$ and $g: \mathbb{Y}\,|\, \mathbb{B}$ be meta processes (v-profunctors).

Let $\mathcal{Q}(\mathbb{X,Y})$ and $\mathcal{R}(\mathbb{A,B})$ be meta relations (h-profunctors).

A meta inference, or double profunctor, is a vertical bimodule of vertical monads $f$ and $g$ in $\mathbb{M}\mathrm{at}\mathbb{C}\mathrm{at}$.

This is a matrix profunctor

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$i(f,g): \mathcal{Q}(\mathbb{X,Y})\, |\, \mathcal{R}(\mathbb{A,B})$

with matrix transformations

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left (pre)composition

$\circ_f: f\otimes i\Rightarrow i$

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right (post)composition

$\circ_g: i\otimes g\Rightarrow i$

so the associators and unitors of $\mathcal{Q,R}$ are natural with respect to $i$

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center associator coherence

$\mathbb{X}\circ (\mathcal{Q}\circ \mathbb{Y}) \rightrightarrows (\mathbb{A}\circ \mathcal{R})\circ \mathbb{B}$

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left associator coherence

$(\mathbb{X}\circ \mathbb{X})\circ \mathcal{Q}\rightrightarrows \mathbb{A}\circ (\mathbb{A}\circ \mathcal{R})$

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right associator coherence

$\mathcal{Q}\circ (\mathbb{Y\circ Y})\rightrightarrows (\mathcal{R}\circ \mathbb{B})\circ \mathbb{B}$

and the unitors are natural with respect to $i$.