Showing Cycle is NL-complete?
Let us reduce direct connectivity to CYCLE. We are given a directed graph $G$ and two vertices $s,t$. Suppose that $G$ contains $n$ vertices. We create $n$ copies of the vertices of $G$. For each edge $x \to y$ in $G$, we add an edge from the $i$th copy of $x$ to the $(i+1)$th copy of $y$. We also add an edge from the $n$th copy of $t$ to the first copy of $s$. The new graph contains a cycle iff $t$ is reachable from $s$ in $G$.

We can construct the new graph from the original graph in logspace. Hence this is a logspace reduction, which shows that CYCLE is NL-hard.