Exact meaning of $2^{\mathcal{O}(f(n))}$

In Sipser’s Introduction to the Theory of Computation he uses the notation $ 2^{\mathcal{O}(f(n))}$ to denote some asymptotic running time.

For example he says that the running time of a single-tape deterministic turing machine simulating a multi-tape non-deterministic turing machine is

$ \mathcal{O}(t(n)b^{t(n)})=2^{\mathcal{O}(t(n))}$ where $ b$ is the maximal number of options in the transtition function.

I was wondering if someone can clarify the exact definition of this for me:

My assumption is that if $ g(n)=2^{\mathcal{O}(f(n))}$ then there exists $ N \in \mathbb{Z}$ and $ c \in \mathbb{R}$ s.t.

$ g(n) \leq 2^{cf(n)}=2^{f(n)^c}$ for all $ n>N$ .

Thank you