This is an artificial answer, I'm looking for something more natural.

In this paper, T. Teruya introduced the notion of normal intermediate subfactors, generalizing exactly the notion of normal subgroups (see the post Jordan-Hölder theorem for subfactors 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.

This is an artificial answer, I'm looking for something more natural.

In this paper, T. Teruya introduced the notion of normal intermediate subfactors, generalizing exactly the notion of normal subgroups (see the post Jordan-Hölder theorem for subfactors 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.