If I understand you correctly, you want to check if values exists in lists,

Since you have a unique name, it's best to leverage that uniqueness in combination with a map,
you start of by adding all the values in your lists to 2 maps.

    Map<String, Obj__c> objMap1 = new Map<String, Obj__c>();
    Map<String, Obj__c> objMap2 = new Map<String, Obj__c>();
    
    // List 1
    for(Obj__c o : list1)
    {
    	objMap1.put(o.UniqueName__c, o);
    }
    
    // List 2
    for(Obj__c o : list2)
    {
    	objMap2.put(o.UniqueName__c, o);
    }
    
    // Now you can easily check if that value is in your map by doing.
    objMap1.containsKey('WhateverYouWantToCompare');
    // or
    objMap2.containsKey(newObj.UniqueName__c);

... do your logic