Learn Finnish: Minä luin kirjaa, kun hän sai kirjahyllyn kootuksi. - I was reading a book when he got the bookshelf assembled.

This is the 'achievement' construction using the verb saada (to get) plus a passive past participle in the translative case (-ksi). It means 'to manage to do something' or 'to get something done.' Here, hän sai kirjahyllyn kootuksi means 'he managed to assemble the bookshelf' or 'he got the bookshelf assembled.'

Why is kirjaa in the partitive case, but kirjahyllyn is in the genitive case?

This perfectly highlights how Finnish handles ongoing versus completed actions. Luin kirjaa uses the partitive because the reading was an ongoing, incomplete process (I was reading a book). On the other hand, kirjahyllyn takes the genitive (-n) because the act of assembling the bookshelf was a fully completed achievement (he got it assembled).

How is the word kootuksi formed?

It comes from the verb koota (to assemble, to gather). First, you form the passive past participle, which is koottu (assembled). Then, you add the translative case ending -ksi, giving you kootuksi. Because koota is a type 4 verb, remember that its passive past participle takes a double -tt-.

Could I use the partitive case instead and say sai koottua?