AWS S3/Lambda Copy object upon upload automagically?

We have multiple buckets that are used by our clients. A client uploads a single file (random filename) to their bucket, and we then visit that bucket and copy it to our own bucket for processing. Basically, this: http://ift.tt/1ESClFm

How could I automate this? I note lambda has s3 object event support so wonder if I can use this somehow?

Paul.