For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of depth $0$ or $2$ correspond to the projections $p_i = I_{H_i}$ on the $2$-boxes space $\mathcal{P}_2(N \subset M) = \bigoplus_{i \in I} End(H_i) $ as an algebra.

*Warning*: $I$ is the index set for the simple objects of depth $0$ or $2$, but there can have others.

On a $2$-boxes space, there is a structure of coproduct $a∗b$ (defined for example [here][1] p4).

**Question**: How compute the coproduct $p_j * p_j$ from the fusion $X_i \boxtimes X_j$?

My guess is that $p_i * p_j \sim \sum_{k \in K} p_k$ with $K=\{ k \in I \ \vert \ X_k \le X_i \boxtimes X_j \}$ (*truncated fusion*).

Is it true? What's the exact formula? How prove it?

Else, if there is a minimal projection $u \preceq p_i∗p_j$ such that its central support $Z(u)=p_k$, but $p_k \not \preceq p_i∗p_j$, is it still true that $X_k$ appears as a summand of $X_i ⊠ X_j$?

In other words, is it true that $Z(p_i * p_j) \le \sum_{k \in K} p_k$? Is it an equality? (*weak truncated fusion*)