Want each client to set a parameter on network object once its created

So our master client creates a network object. There is one parameter in this network object which I want to change for each client depending on the state of that particular client. Initially I tried putting the check in the Start() method but obviously that start method is only run by whoever initially instantiates the network object.

Is there a callback for some other method that every single client will run as soon as it receives notification that a new network object has been created?

I guess alternatively I could try putting something that is run the first time the Update() method is called but that seems really messy. Any advice appreciated. Thanks.