Hp laserjet cp5225 cyan doesn't print

If a service person can do it you should be able to I think. See link below for assistance/support for your device
HP Color LaserJet Professional CP5225 Printer series

No.

Very few printers will print if they're out of toner, even if the colour toner is exhausted and the print job is only black. Replace the empty toner and if this is the only problem, the printer will continue printing.

the transfer unit shold to replace

In the settings menu there should be a economy mode which will lighten up the print.

HP has a list of solutions for print quality problems. Have you tried all the solutions suggested here?

Troubleshoot problem for HP Color LaserJet Professional CP5225 Printer...

There were also a couple Mac specific solutions available (links look the same but are different). Both solutions are about prints being different than expected.

Troubleshoot problem for HP Color LaserJet Professional CP5225 Printer...

Troubleshoot problem for HP Color LaserJet Professional CP5225 Printer...

I am an HP employee.

need to replace those colour cartridges

If your printer is missing a color it could be caused by (off the top of my head) 3 possibilities.
1. High voltage board
2. laser/scanner assembly for that color
3. DC controller might need a firmware upgrade, www.hp.com
Run the PQ diagnostics, check the cyan color
Run the Cartridge Disable diagnostic. Remove all cartridges and take the black cartridge and insert it where the cyan normally is. If the black prints in the cyan slot, your cyan cartridges are bad. If the black doesn't work then go to #3 above. It won't cost any money to upgrade the firmware. If it still doesn't work call in a printer tech.
At this point you may need a printer tech.

You can find the user's guide here which lists all the parts and supplies. Here's the cartridge information from the manual:

• HP Color LaserJet black print cartridge Standard black cartridge CE260A
• HP Color LaserJet black print cartridge High-capacity black cartridge for CLJ4525 models only CE260X
• HP Color LaserJet cyan print cartridge Cyan cartridge CE261A
• HP Color LaserJet yellow print cartridge Yellow cartridge CE262A
• HP Color LaserJet magenta print cartridge Magenta cartridge CE263A

You can use the HP numbers to look for original or after-market replacements.

HP Support & Drivers

Check the black cover on the pick up assembly under the cyan cartridge. If it is only a fraction too high cyan will not print. Make sure it is clipped in place correctly.