= Solution
Write $T=g(Y)$. Because $T$ is determined by $Y$, the <chain rule for mutual information> gives
$$
I(X;Y)=I(X;Y,T)=I(X;T)+I(X;Y\mid T)\geq I(X;T).
$$
This is the <data processing inequality>. It holds for every input <probability distribution>. Maximizing each side over that same set of input <probability distributions> gives \b[$C_{g(Y)}\leq C_Y$]; processing cannot increase <channel capacity>.
A simple sufficient condition for equality is that \b[$g$ be <injective> on the reachable output <alphabet>]. Then $Y$ can be recovered from $T$, giving $H(Y\mid T)=0$ and hence $I(X;Y\mid T)=0$ for every input law. The input that achieves the original <channel capacity> therefore achieves it after processing as well. Values of $g$ outside reachable outputs are irrelevant.
<Injectivity> is sufficient rather than necessary. More precisely, for <finite alphabets> <equality in capacity data processing> holds exactly when there is an original capacity-achieving input law for which $X$ and $Y$ have <conditional independence> given $T$. Sufficiency follows from the displayed gap identity. For necessity, maximize $I(X;T)$ at an input law $p_*$; if its maximum equals $C_Y$, then $C_Y=I_{p_*}(X;T)\leq I_{p_*}(X;Y)\leq C_Y$, so both inequalities are equalities. Processing may discard output details unrelated to the input without losing <channel capacity>.
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