= Solution
After the first <quantum channel>, denote the joint reference-output state by $\omega_{RB_1}$. Dilate the second <CPTP map> as an <linear isometry> $B_1\to B_2E_2$, with the resulting state denoted $\tau$. <Linear isometries> preserve nonzero eigenvalues, both globally and on the transformed output, giving
$$
I_c(\Phi_1,\rho)=S(B_2E_2)_\tau-S(RB_2E_2)_\tau.
$$
Tracing out $E_2$ implements the second channel, so
$$
I_c(\Phi_2\circ\Phi_1,\rho)=S(B_2)_\tau-S(RB_2)_\tau.
$$
The <Strong subadditivity of Von Neumann entropy> for $R,B_2,E_2$ gives $S(RB_2)+S(B_2E_2)\geq S(B_2)+S(RB_2E_2)$. Rearranging proves the <data-processing inequality for coherent information>:
$$
\boxed{I_c(\Phi_1,\rho)\geq I_c(\Phi_2\circ\Phi_1,\rho)}.
$$
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