When uploading the file, add the email address or other identifier to the object as Object User-Defined Metadata.
When uploading an object, you can also assign metadata to the object. You provide this optional information as a name-value (key-value) pair when you send a PUT or POST request to create the object. When uploading objects using the REST API the optional user-defined metadata names must begin with "x-amz-meta-" to distinguish them from other HTTP headers. When you retrieve the object using the REST API, this prefix is returned. When uploading objects using the SOAP API, the prefix is not required. When you retrieve the object using the SOAP API, the prefix is removed, regardless of which API you used to upload the object.
When the Lambda function completes the file processing, it can read that same metadata, and send an appropriate notification to the user.