Let $(X,x_0)$ be a based retract of $(Y,y_0)$ with retraction $R$ and inclusion $\iota$ such that $$R\circ\iota=\mathrm{id}_X.$$
Show that there is a short exact sequence $$1\rightarrow N\rightarrow\pi_1(Y,y_0)\xrightarrow{R_*}\pi_1(X,x_0)\rightarrow 1,$$ where $N:=\ker(R_*)$.
So I thought it suffices to show that $R_*$ is surjective. We have $$R_*\circ\iota_*=\mathrm{id}_{\pi_1(X,x_0)}$$ But this already implies that $R_*$ is surjective and since $N=\ker(R_*)$, the above is an exact sequence.
This seems too simple to me, did I miss something?