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.

* 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.