Valter wrote:

> Presumably the secretary of a department works in that department, so

> $\mathrm{WorksIn} \circ \mathrm{Secretary} = 1_{\mathrm{Department}},$

Yes, this is one of the equations I had in mind!

> $\mathrm{WorksIn} \circ \mathrm{Manager} = \mathrm{WorksIn}$

Yes, that's the other one! Though as you note, we can imagine exceptions.