The function of the warm up is to avoid injury and prepare your body to climb. "The session" is what you've come to achieve. This might be to simply enjoy yourself and relax after work,  The function of the warm down is to help your body recover from the workout. 
The Warm Up: Devoting 20 to 30 minutes to warming up can make all the difference to the quality of any climbing session. Warming up will relax you and allow you to focus on climbing, it will help you avoid injuries (a warm, elastic, stretched muscle is less likely to be damaged than a cold, stiff one) and last but not least it will prepare your body to climb well. 
Warming up is a very personal thing. The length and structure of the warm-up will vary from person to person. It is important to learn what is best for you. You should know how many routes of what grade your body needs to do before it is ready to pull as hard as it can without breaking. Generally speaking, the fitter you get and the older you get the longer the warm-up needs to be. Some professional climbers may spend as much as two hours warming up. 
Warming up has three parts: 1) an aerobic component that increases blood flow and literally warms your large muscle groups; 2) a stretching component that ensures your limbs can obtain their full range of motion and; 3) a more specific warm up on easy routes. This warms up the specific muscles stressed while climbing it also wakes up your coordination and body awareness. 
Warming Down - After your climbing session it is important to spend 10-15 minutes warming down. The aim is to relax and stretch the stressed muscles to assist in the recovery process. Complete a number of easy routes and repeat your stretching routine.
 Have fun and dont forget to warm up and down.