nLab tangent spaces of mapping spaces

Idea

As shown in manifold structure of mapping spaces, the space of smooth maps from a sequentially compact Frölicher space (or diffeological or Chen space) in to a smooth manifold is again a smooth manifold. Its tangent space is straightforward to identify. A tangent vector is an infinitesimal deviation of a smooth map; that is, it defines a direction in which to deform that map. As a smooth map is determined by its values at points, when deforming a smooth map it is enough to explain how to deform each point. Thus a tangent vector at $\alpha \colon S \to M$ defines, for each $p \in S$, a tangent vector at $\alpha(p) \in M$. Thus we obtain a map $S \to T M$. It is not unbelievable that this map is again smooth, whence we have:

$$T C^\infty(S,M) \to C^\infty(S, T M).$$

The claim of this page is that this is a diffeomorphism, and further a vector bundle isomorphism, where each is considered a vector bundle over $C^\infty(S, M)$.