Let $\mathbb{X_0,X_1,A_0,A_1}$ be logics (bifibrant double categories),

$f_0:\mathbb{X_0}\,|\, \mathbb{A}_0$ and $f_1: \mathbb{X_1}\,|\, \mathbb{A_1}$ be meta processes (v-profunctors),

$[\![\mathbb{X}]\!]:\mathbb{X_0\to X_1}$ and $[\![\mathbb{A}]\!]:\mathbb{A_0\to A_1}$ be flow types (double functors).

A flow process, or vertical transformation,

is a matrix transformation $[\![f]\!]:f_0\Rightarrow f_1$

3.png

which coheres with the joins of the double functors.

4-comp.png

4-unit.png

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