The dihedral $D_4$ group contains the subgroup
$$H=\langle r^2,t\rangle=\{e, r^2,t,tr^2 \}$$
which is normal. A proof is offered right here, based on the fact that $H$ is the kernel of a homomorphism, $\ker(\phi)=H:$
$$\begin{align} \phi &: D_4\to\mathbb Z_2\\ & r \mapsto 1\\ & t \mapsto 0 \end{align}$$
I understand why by the fundamental theorem of homomorphisms, given this homomorphism, $H$ is normal. So the fact that it is a normal subgroup is not the issue - in fact, $[D_4:H]=2,$ also proving that $H$ is normal.
But the question is how are the arrows in the map above generated from $r\mapsto 1$ and $t\mapsto 0.$