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

A meta process or vertical profunctor is a vertical monad between pseudomonads $\mathbb{X}$ and $\mathbb{A}$ in MatCat.

This is

a profunctor $\underline{f}:\underline{\mathbb{X}} \,|\, \underline{\mathbb{A}}$ of processes and

a matrix profunctor of inferences

2.png

$f(\underline{f},\underline{f}): \mathbb{X}(\underline{\mathbb{X}},\underline{\mathbb{X}})\, |\, \mathbb{A}(\underline{\mathbb{A}}, \underline{\mathbb{A}})$

with matrix transformations

3-comp.png

parallel composition

$\circ: f\otimes f\Rightarrow f$

3-unit.png

parallel identity

$\mathrm{id}: \underline{f}\Rightarrow f$

so the associators and unitors of $\mathbb{X,A}$ are natural with respect to $f$.

4-assoc.png

associator coherence

$(\mathbb{X} \circ \mathbb{X}) \circ \mathbb{X} \rightrightarrows \mathbb{A}\circ (\mathbb{A}\circ \mathbb{A})$

4-lunit.png

left unitor coherence

$\mathrm{id}.\underline{\mathbb{X}}\circ \mathbb{X} \rightrightarrows \mathrm{id}.\underline{\mathbb{A}}\circ \mathbb{A}$

4-runit.png

right unitor coherence

$\mathbb{X}\circ \mathrm{id}.\underline{\mathbb{X}}\rightrightarrows \mathbb{A}\circ \mathrm{id}.\underline{\mathbb{A}}$