On a course, when shift systems were being introduced, the lector said that "if the shift of symbols sequence reminds you Turing machine, then it is a very correct association":
$\sigma(\ldots, x_{-1}, x_0, x_1, \ldots) = (\ldots, x_0, x_1, x_2, \ldots)$
I asked him about concretes but he gave me vague answers that seem to only refer to the general notion of universality that can be proved by reducing a system to Turing machine.
So he did not confirm his suggestion, that shifting sequence of symbols is analogous to moving the tape in Turing machine. But I am confused. Can this be clarified?