= Solution
Put $\theta=\operatorname{cf}(\delta)$. There is a strictly increasing cofinal map $a:\theta\to\delta$. If $\operatorname{cf}(\theta)=\nu<\theta$, choose a cofinal map $b:\nu\to\theta$. Then $a\circ b$ is cofinal in $\delta$, contradicting the definition of $\theta$ as its least cofinal <order type>. Since always $\operatorname{cf}(\theta)\le\theta$, we obtain
$$
\boxed{\operatorname{cf}(\operatorname{cf}(\delta))=\operatorname{cf}(\delta).}
$$
For a nonzero <limit ordinal> this is an infinite <regular cardinal>. If the convention includes zero among <limit ordinals>, its <cofinality> is zero and the identity is immediate there.
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