Lemma: Local isometric dictionaries preserve the task

Separate isometric embeddings of $R$ and $B$ preserve ${\epsilon }_{\mathrm{rec}}$ and ${\delta }_{\mathrm{full}}$. They do not exchange accessible and inaccessible outputs.

Proof. If $J:H\to H^{\prime }$ is an isometry, its channel has the CPTP retraction

where ${\tau }_H$ is any fixed state. Embedding candidate decoders or replacement states gives one inequality. Applying the retraction and contraction gives the reverse inequality. The two embeddings act on their respective local factors only.