The "for X to Y" pattern is a bit idiomatic (unusual, one-off). You could think of it like a special kind of prepositional phrase (the preposition for is often used to give reasons—e.g. "I bought flowers for my mother's birthday").
If you want to leave it there, that's fine. If you're curious about why we think this pattern is weird, read on.
In -ing and -ed clauses, you can use an implicit or explicit subject
With an implicit subject:
- A plank of wood lay by the track, roughly fashioned in the shape of an arrow.
(The plank of wood is carried through from the main clause to the supporting clause.)
With an explicit subject:
- Aisha was now finishing the spice cake, her eyes firmly cast down.
(The eyes are new, so have to be included in the supporting clause.)
Fairly straightforward—if the subject is the same as the main clause, we leave it out, and if it's different we just add it in.
But for whatever reason, it works slightly differently for infinitive clauses
Here's a back-to-back example of an infinitive clause with an implicit and explicit subject.
With an implicit subject:
- Lorraine had to skip a bit to catch up.
(Lorraine is carried through from the main clause to the supporting clause.)
With an explicit subject:
- Lorraine had to slow down for me to catch up.
(The narrator is new, so has to be added explicitly to the supporting clause, but we have to use this special "for" because of the weird way that infinitive clauses work.)
We're not sure why infinitive clauses work like this. They just do 🤷