HELP!! How to using this scheme Arduino---router---internet---router---Arduino

Using Arduino Networking, Protocols, and Devices
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With the ethernet shield, the Arduino can act as a client, making requests to servers, or it can act as a server, handling requests from clients.

Which role should each of your Arduinos perform? When you have that sorted out, making the client make a GET request, to have a server turn on an LED, is trivial, as is handling the request on the server.