Homotopy groups of connected components

Question

Let $f:X\to Y$ be a (surjective if it helps) map of simplicial commutative monoids such that the induced map on homotopy groups $\pi_n(X,1)\to \pi_n(Y,1)$ is an isomorphism for all $n\geq 0$. Let $x\in X$. Is $\pi_n(X,x)\to \pi_n(X,f(x))$ necessarily an isomorphism? (I don't even know if homotopy groups with basepoint in different connected components of a simplicial commutative monoid can be non-isomorphic).

Answer 1

No. For instance, let $A$ be any connected simplicial commutative semigroup, and let $X=A\sqcup\{*\}$ with the monoid structure such that $*$ is the identity. Let $B$ be another connected simplicial commutative semigroup and $Y=B\sqcup\{*\}$ similarly. Then any (surjective) homomorphism $A\to B$ induces a (surjective) homomorphism $f:X\to Y$, which is an isomorphism on all homotopy groups based at the identity. However, $f$ is not an isomorphism on homotopy groups based vertices in $A$ unless the original map $A\to B$ was a weak equivalence.