Rice's Theorem for Turing machine with fixed output
A word $w$ belongs to $L_5$ if for all $x \in \{0,1\}^*$ it is the case that $M_w(w) = x$. In particular, if $w \in L_5$ then $M_w(w) = 0$ and $M_w(w) = 1$, which can't both be true. So no word belongs to $L_5$.