Prove the existence of $F \in \Lambda$ such that $Fx = xF$ for arbitrary variable $x \in \Lambda$

Question

I recently met the problem as indicated in the title: find an $F \in \Lambda$ such that $Fx = xF$ for arbitrary variable $x \in \Lambda$. I am not only seeking a solution, but also a systematic way to think about such problems. Could someone help?

Answer 1

Finally I got the solution after reviewing materials from Harvard. From \begin{equation} F = \lambda x. xF, \end{equation} we may define \begin{equation} F^{\prime} = \lambda t. \lambda x. x\left(tt\right). \end{equation} Then \begin{equation} F = F^{\prime}F^{\prime}. \end{equation}

Indeed, without reading the materials, I couldn't have solved this problem.