I am trying to prove that the following operator
\begin{align*} F: C^2 ([0,T], \mathbb{R}) &\rightarrow C ([0,T], \mathbb{R}) \\ f &\mapsto f'' + p \circ f \end{align*}
is $C^{\infty}$, where $p: \mathbb{R} \rightarrow \mathbb{R}$ is a $C^{\infty}$ function, but I haven't managed to do it. I tried to write $F$ as a composition of functions, but the map $f \mapsto f''$ isn't $C^\infty$, is it?