= Solution
A <discrete memoryless channel> has finite input alphabet $\mathcal A$, finite output alphabet $\mathcal B$, and transition probabilities $P(y\mid x)$; successive outputs are conditionally independent given the corresponding inputs. The <Shannon second coding theorem> states that rates below
$$
C=\max_{P_X}I(X;Y)
$$
admit block codes with error probability tending to zero, whereas rates above $C$ cannot have vanishing error.
For any joint input law of $(X_1,X_2)$, conditional independence of the product channel gives
$$
\begin{aligned}
I(X_1,X_2;Y_1,Y_2)
&=H(Y_1,Y_2)-H(Y_1,Y_2\mid X_1,X_2)\\
&\leq H(Y_1)+H(Y_2)-H(Y_1\mid X_1)-H(Y_2\mid X_2)\\
&=I(X_1;Y_1)+I(X_2;Y_2)
\leq C_1+C_2.
\end{aligned}
$$
Choose $X_1$ and $X_2$ independently with capacity-achieving input laws for their respective channels. Then the inequality becomes equality, so the product-channel capacity is
$$
\boxed{C_1+C_2}.
$$
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