customization of the forgot password email

The answer to your question is Custom message Lambda trigger. Amazon Cognito invokes trigger before it sends an email or phone verification message or a multi-factor authentication (MFA) code. You can customize the message dynamically with your custom message trigger. The below doc describes the various ways you can use a custom message lambda. https://docs.aws.amazon.com/cognito/latest/developerguide/user-pool-lambda-custom-message.html