I am a physics undergraduate reading through section 2.1 of Sakurai's Modern Quantum Mechanics (3ed).

I have been having a hard time parsing the notation of the section since the same symbols seem to be used for different mathematical manipulations. In particular,

Q: How do I write the action of the unitary time-evolution operator $\mathcal{U}(t_0, t)$ on a state ket $\alpha$ precisely? To my understanding, the content of the section does not make sense if you treat $\mathcal{U}(t_0, t)$ as a function of $t$. Instead, it seems one should treat the operator like
$$ \mathcal{U}(t_0, t') |_t \cdot | \alpha \rangle.$$

In other words, that when one acts with the unitary time-evolution operator, one is implicitly evaluating the operator at a particular value $t$ and then acting on the state ket. 

Is this accurate?