Let $M$ and $N$ be finite dimensional compact manifolds. Denote the set of $C^1$-maps from $M$ to $N$ by $C^1(M,N)$.
To my knowledge, the following is true: If we equip $C^1(M,N)$ with the $C^1$-topology, the topological space $C^1(M,N)$ has the structure of a smooth ($C^\infty$) Banach manifold.
A chart around $f\in C^1(M,N)$ is usually given by
$\varphi_f\colon U_f\rightarrow V_f, \hspace{3em} g\mapsto (p\mapsto exp^{-1}_{f(p)}g(p))$
Here, $U_f\subset C^1(M,N)$ and $V_f\subset \Gamma_{C^1}(f^*TN)$ are suitable open sets and $exp$ is the exponential map of $N$ with respect to a fixed Riemannian metric on $N$.
I want to understand if these charts turn $C^1(M,N)$ into a smooth Banach manifold, i.e. are the maps
$\varphi_f\circ\varphi_g^{-1}\colon \varphi_g(U_f\cap U_g)\rightarrow \varphi_f(U_f\cap U_g), \hspace{3em} s\mapsto (p\mapsto(exp^{-1}_{f(x)}\circ exp^N_{g(x)})s(x))$
smooth for $f,g\in C^1(M,N)$? Are the $\varphi_f\circ\varphi_g^{-1}$ only smooth for $f,g\in C^\infty(M,N)$? So:
Question 1: Are the $\varphi_f\circ\varphi_g^{-1}$ smooth for $f,g\in C^1(M,N)$?
If not,
Question 2: Are the $\varphi_f\circ\varphi_g^{-1}$ smooth for $f,g\in C^\infty(M,N)$?
I also would appreciate any kind of reference where these questions are treated or advices how to prove the above. Thanks in advance.