[FOM] Question about cardinal collapse

[Colin McLarty]

I have a question about cardinal collapse in set theory.  Let me make 
sure I have the standard definiton.  I take cardinal collapse in an 
extension of a universe of sets to mean: some two sets not isomorphic 
in the original universe are isomorphic in the extension.

Unless I have badly misunderstood, it implies the following condition:  
some two sets S and S' in the original model gain at least one new 
function f:S-->S' in the extension, that is at least one function which 
does not exist in the original model.  

Are those two conditions equivalent?  Does every extension of a 
universe of sets which adds new functions necessarily make some 
originally non-isomorphic sets isomorphic?  If not, is there a standard 
name in set theory for the 
condition of adding new functions? 

For all I have found so far, there may be some terribly easy way to see 
that adding a new function (between sets in the original universe) to a 
universe of sets always adeds at least one isomorphism between sets 
that were not isomorphic in the original universe.  Or there may be 
some well known forcing extension, for example, that does add new 
functions without cardinal collapse.  Can anyone here tell me?  

thanks.  Colin