A fixed point of a function $f$ is an element $a$ such that $f(a)=a$. The condition is equivalent to the property that $f$ maps every number to a fixed point. Counting by the number of fixed points of $f$, the total number of such functions is $$\begin{aligned} \sum_{k=1}^{5}\binom{5}{k} k^{5-k} & =1 \cdot\left(5^{0}\right)+5 \cdot\left(1^{4}+4^{1}\right)+10 \cdot\left(2^{3}+3^{2}\right) \\ & =1+25+10 \cdot 17 \\ & =196 \end{aligned}$$