So Sakurai in their QM book defines the propagator in wave mechanics as:
$$K(x'',t;x',t_0)=\sum_{a'}\langle x''\vert a'\rangle \langle a'\vert x'\rangle \exp\left[\dfrac{-iE_{a'}(t-t_0)}{\hbar}\right].$$
Since the exponential terms are the eigenvalues of $\vert a'\rangle$, you could write this expression as:
\begin{align} K(x'',t;x',t_0)&=\sum_{a'}\langle x''\vert \exp\left[\dfrac{-iH(t-t_0)}{\hbar}\right] \vert a'\rangle \langle a'\vert x'\rangle \\ &= \sum_{a'}\langle x''\vert U(t,t_0) \vert a'\rangle \langle a'\vert x'\rangle \end{align} with $U(t,t_0)$ being the time evolution operator. Then, just eliminating the completeness relation you'd have:
$$\langle x''\vert U(t,t_0)\vert x'\rangle\,,$$
which are the matrix elements of the time evolution operator in the position basis. Is this correct or did I miss something? Are the two concepts describing the same thing?