reference request – Proving that $C_S^{infty}(M,N)$ is a Baire space

I have been reading Hirsch’s Differential topology and I am sure a lot of you know this book as a lot of typos. I believe one of them is the proof that $C_S^{infty}(M,N)$ is a Baire space. I don’t think his proof works, but I hope this is a true fact since it is used throughout the text. Now when he proves that $C_S^r(M,N)$ is a Baire space , for $0 leq r<infty$, we just use a continuous function $J^r:C^r_S(M,N)rightarrow C^0(M,J^r(M,N))$ such that the image is a weakly closed subset of $C^0(M,J^r(M,N))$ and now since $J^r(M,N)$ is complete $M$ is a manifold and $J^r$ is continuous we get that $C_S^r(M,N)$ is a Baire space. Now I think he wants to do the same type of argument for $J^{infty}:C^{infty}(M,N)rightarrow C^0(M,J^{infty}(M,N))$ but here we don’t have the fact that the function is continuous. So my question is if anyone knows a proof or a reference for seeing that $C_S^{infty}(M,N)$ will in fact be a Baire space? Thanks in advance.