= Solution
Both output states have the same reference marginal, and $S(R)=S(\rho)$. Expand <quantum mutual information> to obtain the <mutual information and coherent information identity>
$$
I(R:B_1)_\sigma=S(\rho)+I_c(\Lambda_1,\rho),\qquad
I(R:B_2)_\omega=S(\rho)+I_c(\Lambda_2\circ\Lambda_1,\rho).
$$
The input-entropy term is identical in the two expressions. Subtract them and use the preceding <data-processing inequality for coherent information>:
$$
\boxed{I(R:B_1)_\sigma-I(R:B_2)_\omega
=I_c(\Lambda_1,\rho)-I_c(\Lambda_2\circ\Lambda_1,\rho)\geq0.}
$$
Thus the final <quantum channel> cannot increase the reference-output <quantum mutual information>, as required by <data processing for quantum mutual information>.
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