The construction goes like this:
(comma) "um" + ⟨the verb's object(s)⟩ + "zu" + ⟨infinite form of the verb⟩.
Verbs can have no object at all, or one or some of these kinds of objects:
- no object
Ich schlafe.
- accusative object
Ich sehe den Mann.
- dative object
Du ähnelst meinem Bruder.
- genitive object
Wir gedenken der Toten.
- prepositional object
consists of a preposition and an inner object (which can be accusative or dative, depending on the preposition)
Er achtet auf den Verkehr.
Er berichtet über das Ereignis.
Er stimmt für die Erhöhung.
Er protestiert gegen die Erhöhung.
Er schreibt an seinen Onkel.
Er beginnt mit der Prozedur.
Es besteht aus mehreren Teilen.
Es fehlt an Munition.
Er fragt nach dem Weg.
Er bittet um einen Gefallen.
Any prepositional object consist of a preposition at the beginning and an inner object, that can be an accusative or a dative object (the case is depending on the preposition). The preposition um is one of these prepositions that can stand at the beginning of the verbs prepositional object.
Example:
The verb bitten needs a prepositional object, that must start with the preposition um:
Simon bittet um Aufschub.
Simon asks for a postponement.
If you use this verb in an um zu construction, it looks like this:
Simon ging zum Chef, um um Aufschub zu bitten.
Simon went to the boss to ask for a postponement.
The first um is the um from the um zu construction (um zu bitten = to ask). The second um is the preposition from the prepositional object (um Aufschub = for a postponement).
Many verbs need two (or even more) objects, and one of them can be a prepositional object:
Klaus bittet seinen Vater um einen Gefallen.
Klaus asks his father for a favor.
Here »seinen Vater« is an accusative object of the verb bitten and »um einen Gefallen« is a prepositional object of the same verb. Let's build an um zu construction with these parts:
Klaus fuhr nach Hause, um seinen Vater um einen Gefallen zu bitten.
Klaus drove home to ask his father for a favor.
A verb that has an accusative object is called a transitive verb. But German has also a class of verbs, where this accusative object mandatorily must be a reflexive pronoun. We call these verbs »reflexive verbs«. And among the reflexive verbs there are also some that in addition to the reflexive accusative object also can have a second object, which often must be a prepositional object:
Ich kümmere mich um meine Frau.
I take care of my wife.
Ich bleibe daheim, um mich um meine Frau zu kümmern.
I stay at home to take care of my wife.
Here the word »mich« is the singular 1st person version of the mandatory reflexive pronoun that must exist when the verb »kümmern« is used. It refers to the subject (»ich«) which indicates the caretaker. The part »um meine Frau« is the prepositional object, that indicates the person being cared for.
But you also can replace »meine Frau« by a personal pronoun if she was mentioned earlier:
Ich bleibe bei meiner Frau, um mich um sie zu kümmern.
I stay with my wife to take care of her.