I have to imagine there's a way to amend the form that communicates that the client was serviced for a month; partially by your co-worker, partially by you.
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If there isn't, then I'm not sure if it would be lying, but for your own sake and protection it would be best that it communicated that you were not responsible for the entirety of the service (e.g. suppose a complaint is filed for this time period and it fell into your co-workers "half" of the service).
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If there's not a way to amend it, maybe just put a note by your signature with the beginning and end date of the service you provided.