I haven't personally tried the two devices together, but Cisco POE switches and injectors have been infamous for failing to power IP cameras. They typically don't conform to the 802.3af/at POE standards but use their own proprietary standard, whereas your camera does conform to 802.3af for its power source.
I've just had a quick scan online and it seems that the Cisco Air-Pwrinj3 doesn't conform to the standard.
Whilst this isn't a guaranteed indicator that the injector is at fault, we've seen this exact scenario a lot in the past, and far less often is the camera itself at fault.
You could purchase an injector or switch that conforms to the standard and test to rule out the injector, or if you have a 12V DC PSU kicking around, you could try powering the camera with this instead. Make sure you try different cabling too, preferably with network cable you know is functioning fine, in order to rule out a cable issue.