Why is this extension of composite fields a Galois extension?

Question

Let $k$ be a finite field and let $L/k$ be a a finitely generated field extension. Furthermore, let $K$ be a finite Galois extension of $L$ and let $k^\prime$ denote a finite intermediate field $k\subset k^\prime\subset K$ which is algebraically closed in $K$.
Here comes my question: Why is the extension of the composite fields $K\cdot k^\prime/L\cdot k^\prime$ a finite Galois extension? Thank you for your help!