Lemma: Continuation identity

For a channel $\Lambda :L({\mathcal{H}}_B)\to L({\mathcal{H}}_{B^{\prime }})$,

where ${\mathcal{N}}_A$ destructively reads $A$ into $R$. Outputs are identified using the canonical wire permutation.

Proof. A channel on $B$ commutes with restriction over $S$ and readout of $A$. Apply lem:copytrace; readout of the already block-diagonal $A$-reduction replaces its basis label by $R$.