This article shows how to synchronize an assignee to a text custom field.
Source side
Outgoing sync
replica.assignee = issue.assignee
Destination side
An assignee is an object with some properties. Each property is of type text. You can sync the following properties: key, email, displayName, username. Add the code below to the Incoming sync to sync Assignee email address.
issue.customFields."Your custom field name".value = replica.assignee?.email
You can replace the .email property with any other assignee object property.
Check User for all available properties. For example, to sync the assignee displayName use this code.
issue.customFields."Your custom field name".value = replica.assignee?.displayName