Showing that a map on a tubular neighbourhood is a diffeomorphism

Let $$S$$ be an embedded submanifold of a smooth manifold $$M$$. Then there exists a tubular neighbourhood $$U$$ of $$S$$ in $$M$$, that is, $$U$$ is the diffeomorphic image of $$\exp$$ restricted to the normal bundle $$NS$$ of $$S$$ of a subset $$V \subseteq \mathcal{E} \cap NS$$, where $$\mathcal{E} \subseteq TM$$ is the domain of the exponential map, of the form $$V = \{(x,v) \in NS : |v| < \delta(x)\}$$ for some positive continuous function $$\delta : S \to \mathbb{R}$$.

Let $$0 \leq t \leq 1$$. Then we can define $$\psi_t : U \to U$$ by $$\psi_t(\exp(x,v)) := \exp(x,tv).$$

Now the book Introduction to Symplectic Topology by McDuff and Salamon claims that this map is a diffeomorphism for $$t > 0$$. I do not quite see why. I mean, this map is injective and smooth, but I fail to see surjectivity or the explicit form of an inverse. Moreover, I guess that this map is a local diffeomorphism by considering its derivative. Can anyone help?

After a quick correspondence with the second author, the proof goes as follows: $$\psi_t$$ is a diffeomorphism onto its image, because $$\psi_t$$ is injective since $$\exp_S$$ is. Moreover, an explicit inverse is given by $$\exp_S(x, tv) \mapsto \exp_S(x, v)$$.