No, Pierre, there's no analogy like that. Running processes **in series** is composing morphisms

$f \colon x \to y , \quad g \colon y \to z$

to get a morphism

$gf \colon y \to z .$

We can do this in any category.

Running processes **in parallel** is tensoring morphisms

$f \colon x \to y, \quad f' \colon x' \to y'$

to get a morphism

$f \otimes f' \colon x \otimes x' \to y \otimes y' .$

We can do this in any [monoidal category](https://en.wikipedia.org/wiki/Monoidal_category).

None of this has anything to do with right and left adjoints!