Please try the Update connection endpoint of the Auth0 management API.
Here is the sample curl command:
curl --location --request PATCH 'https://{{auth0_domain}}/api/v2/connections/<CONNECTION_ID>;' \
--header 'Content-Type: application/json' \
--header 'Authorization: Bearer <MGMT_API_BEARER_TOKEN>
--data-raw '{
"options": {
"fieldsMap": {
"name": "fullName",
"email": "email",
"given_name": "firstName",
"family_name": "lastName"
}
}
}'