This is an artificial answer, I'm looking for something more natural.  
 
In [this paper][1], T. Teruya introduced the notion of normal intermediate subfactors, generalizing exactly the notion of normal subgroups (see the post [Jordan-Hölder theorem for subfactors][2] for more details).  

So we can generalize the group-morphisms to the subfactors as follows :   
A (group-like) morphism for $(A \subset B)$ to $(C \subset D)$ is the data of:   
 
 - a normal intermediate subfactor $(A \subset P \subset  B)$    
 - an intermediate subfactor $(C \subset Q \subset  D)$   
 -  a $W^*$-isomorphism $\phi_l : (A \subset P) \to (Q \subset  D)$ **or** $\phi_r : (P \subset B) \to (C \subset  Q)$    

**Remark**: This notion generalizes by construction the group-morphisms, unfortunately, it's a bit artificial, I would prefer a more natural definition of morphisms, without using 'ad hoc' the notion of normal intermediate subfactors, but such that the kernel of these natural morphisms are exactly the normal intermediate subfactors.  

   


  [1]: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jmsj/1225376618&page=record
  [2]: https://mathoverflow.net/questions/156311/jordan-holder-theorem-for-subfactors