Learn Turkish: Müdür, süiti kâhyaya temizlettirdi. - The manager had the butler get the suite cleaned.

The base verb is temizlemek (to clean). Because the stem ends in a vowel, the first causative suffix is -t, making temizletmek (to have something cleaned). To add the second causative layer, we use the -DIr suffix. Due to consonant mutation after the voiceless t and vowel harmony, it becomes -tir, resulting in temizlettirmek (to have someone get something cleaned). Finally, -di is the simple past tense ending.

Why does kâhyaya take the dative case (-ya)?

When a causative verb takes a direct object (like süiti, which is marked with the accusative case), the person who is ordered to manage the action is marked with the dative case. The manager is directing the action to the butler (kâhya), so it becomes kâhyaya.

Is the person who actually cleans the suite mentioned in this sentence?

No, they are left out. We have the instigator (müdür) and the intermediary (kâhya). The actual cleaner is omitted, which is very common in double causative sentences to avoid getting bogged down in too many noun cases. If you really needed to mention the cleaner, you could use a phrase with tarafından (by), but context usually makes it unnecessary.

Why doesn't the t in süiti soften to a d through consonant mutation?