A category is seen as a 1-dimensional structure of objects and morphisms; yet reasoning in a category consists of 2-dimensional equalities between composites of morphisms.
Every category forms a double category, in fact three double categories, whose squares are commuting squares. Two are known: the arrow double category $\mathbb{\overrightarrow{A}}$ and its opposite $\mathbb{\overleftarrow{A}}$; their modules are fibrations and opfibrations.
Yet $\mathbb{\overrightarrow{A}}$ and $\mathbb{\overleftarrow{A}}$ are not logics; each has only companions or conjoints.
So we define the weave double category $\mathbb{\langle A\rangle}$ to be the coproduct $\mathbb{\overrightarrow{A}}
The weave construction $\langle -\rangle$ applies to categories and functors, profunctors and transformations.
This defines a mapping
from categories and profunctors to logics and metaprocesses
— but it is neither lax nor colax.