Function (a.k.a. “functor”)

Relation (a.k.a. “profunctor”)

Transformation

An inference $i:Q\Rightarrow R$ over processes $\mathrm{f:X\to A, g:Y\to B}$

trans-2.png

with naturality equations: mapping commutes with composition.

trans-3-l0.png

$i(\mathrm{x}\cdot q)$

trans-3-l1.png

$\mathrm{f(x)}\cdot i(q)$

trans-3-r0.png

$i(q\cdot \mathrm{y})$

trans-3-r1.png

$i(q)\cdot \mathrm{g(y)}$