= Solution
Let $T\subseteq(X\times\omega)^{<\omega}$ be a <Suslin representation> of $A$, and let $f:X\to Y$ be injective. Apply $f$ coordinatewise to the first coordinate of every node and put
$$
T_f=\{(f\circ s,t):(s,t)\in T\}\subseteq(Y\times\omega)^{<\omega}.
$$
The injectivity of $f$ ensures that a sequence is a branch of $T_f$ exactly when its first coordinate decodes to a branch of $T$. Hence $p[T_f]=p[T]=A$, proving that every <X-Suslin set> is $Y$-Suslin whenever $X$ injects into $Y$.
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