Should you eat after climbing?

It is worth eating after climbing, but it really depends on the time you spent climbing and at what level of effort you climbed at.

If you spent most of the session sitting or just not climbing, eating right after may not be that important, and you could probably just eat awhile after.

If you feel you spent a lot of energy while climbing and are drained, it may be better to eat after.

It also really depends on your diet. Many people train and climb while fasting, so they would only eat when their next meal is scheduled.
In general, if you are not on any specific diet or regime, you can eat right after or wait a while, it is up to you. If you feel weak, faint, or drained, then you just may want to eat something sooner than later.