 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 putThe 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 $Y$.