In short: I don't know the technical reason, but you can not.
I think many things are not still clear about PoE.
I have tested a TL-POE150S with a UniFi AP, and the Ubiquiti device started blinking and failing (I was not able to connect to the WiFi network) after a few minutes.
I then changed and put its own (included) injector, and everything went fine (I have the device working 24 hours a day).
So, I assume the auto-sensing capability of the injector does not apply to Voltage, and keeps sending 48 VDC, not 24 VDC as the device requires. That is a bad thing when talking about electronics.
Like most Ubiquiti devices as for today (February 2016), they are PoE but not 802.3af (nor 802.3at) compliant (as so, you will not find such words on its docs), but rather Passive PoE.
This is: the device will start (boot) if you send the correct voltage via ethernet cable but will have no voltage negotiation capabilities.
In this case, your TL-POE150S is sending 48 Volts to the UniFi AP, that according to its specifications works with Passive Power over Ethernet (12-24V)
.
Sending 48 Volts (twice the proper voltage) will make the device to become hot, hang, and even become damaged as time goes by.