Let $\mathbb{A_0,A_1,B_0,B_1}$ be logics.

Let $\mathcal{R}_0(\mathbb{A_0,B_0})$ and $\mathcal{R}_1(\mathbb{A_1,B_1})$ be meta relations.

Let $[\![\mathbb{A}]\!]:\mathbb{A_0\to A_1}$ and $[\![\mathbb{B}]\!]:\mathbb{B_0\to B_1}$ be flow types.

A flow relation, or horizontal transformation,

is a matrix functor $[\![\mathcal{R}]\!]: \mathcal{R_0\to R_1}$

2.png

with matrix transformations, left join and right join

3l.png

3r.png

which cohere with the associators

4ac.png

4al.png

4ar.png

and the unitors.

4-unit.png