Sophie! Yes, this "stuff, structure and properties" trinity is exactly what's going on here. We can take any functor \$$f: A \to D\$$ and write it as

* a functor \$$f_1 : A \to B\$$ that **forgets only stuff**: it's essentially surjective and full (but not necessarily faithful)

followed by

* a functor \$$f_2 : B \to C\$$ that **forgets only structure** it's essentially surjective and faithful (but not necessarily full)

followed by

* a functor \$$f_3: C \to D\$$ that **forgets only properties**: it's faithful and full (but not necessarily essentially surjective).

In the case when \$$A\$$ and \$$D\$$ are preorders, so are \$$B\$$ and \$$C\$$, and we are in the situation discussed by Tobias.

(Tobias made everything monoidal, just to spice it up, but it works fine in the non-monoidal situation too.)

If you ever want to dig _really_ deep into this stuff, structure and properties business, read my [Lectures on \$$n\$$-categories and cohomology](http://math.ucr.edu/home/baez/cohomology.pdf). It may be rather stressful, but it explains how this trinity continues when we go to \$$n\$$-categories, and how it's connected to homotopy theory.