Peano categoricity is equivalent to weak konig lemma

Question

Peano categoricity (PC) says that: Every model for second order peano system is isomorphic to standard model. i.e PC says that every peano system such as (A, f, i) is isomorphic to (N, S, 0). Simpson and Yokoyama in 2013 proved that Peano categoricity (PC) is equivalent to weak konig lemma (WKL) over RCA zero. My question is about The set A. If we are in RCA, what A can be?! It is necessarily a subset of N. i.e A can't be uncountable. So this theorem that proved by Simpson and Yokoyama is a weaker version of peano categoricity. i.e they have proved that Peano system is Aleph zero categorical. Am I right?! If I missed sth, please let me know.