**Step 1: Forward ($H \unlhd G \implies H$ is a union of conjugacy classes).** Let $h \in H$. For any $g \in G$, normality gives $ghg^{-1} \in H$, so $\operatorname{ccl}_G(h) = \{ghg^{-1} : g \in G\} \subseteq H$. Therefore: \begin{align*} H = \bigcup_{h \in H} \operatorname{ccl}_G(h). \end{align*} **Step 2: Backward ($H$ is a union of conjugacy classes $\implies H \unlhd G$).** Suppose $H = \bigcup_{h \in I} \operatorname{ccl}_G(h)$ for some index set $I \subseteq H$. For any $a \in H$ and $g \in G$, the element $gag^{-1} \in \operatorname{ccl}_G(a) \subseteq H$. By condition (iii) of the [Equivalent Definitions of Normality](/theorems/787), $H \unlhd G$.