Yes, the map you mention is an isomorphism. I think the main reason people rarely address your specific question in literature is that the technique of the proof is more important than the theorem. But all the main tools are written up in read-to-use form in Hirsch's Differential Topology textbook.
There are two steps to prove the theorem. Step 1 is that all continuous maps can be approximated by (neccessarily) homotopic smooth maps. The 2nd step is that if you have a continuous map that's smooth on a closed subspace (say, a submanifold) then you can approximate it by a (neccessarily) homotopic smooth map which agrees with the initial map on the closed subspace. So that gives you a well-defined inverse to your map $\phi$.
There are two closely-related proofs of this. Both use embeddings and tubular neighbourhoods to turn this into a problem for continuous maps defined on open subsets of euclidean space. And there you either use partitions of unity or a "smoothing operator", which is almost the same idea -- convolution with a bump function.