I am trying to write a trigger to do the following.. 

I have a custom object called Project with the API Name of - pse__Proj__c
I have another custom object called Budget with the API Name of - pse__Budget__c

What I want to do with the trigger is.. 
When a new project is created I want the project name to automatically fill a field on the budget object called project. The project field on the budget object is a lookiup field to the project object and has the API name of pse__Project__c. 

Later on I want to also incorporate the trigger to update a field on the budget object called Amount, that has the API name of 
pse__Amount__c with the value found on the Project object in the field called Prouct Budget MAP with the API name of Product_Budget_MAP__c. That can wait but I do need the first step implemented in order to move forward. 

HERE IS WHAT I HAVE SO FAR... Any help is much appreciated!! 


trigger updateBudgetProject on pse__Budget__c (after insert){
    
    List<pse__Budget__c> BudgetToUpdate = new list<pse__Budget__c>();
    
    for(pse__Budget__c bud : [select Id, pse__Proj__c.Id From pse__Budget__c Where Id IN : trigger.newMap.keyset()]){
        bud.pse__Project__c = bud.pse__Proj__c.name;
        BudgetsToUpdate.add(bud);
       }
    
    if(!BudgetsToUpdate.isEmpty())
        update BudgetsToUpdate;