  We study mapping tori of quasi-autoequivalences $\tau : \mathcal{A} \to
\mathcal{A}$ which induce a free action of $\mathbf{Z}$ on objects. More
precisely, we compute the mapping torus of $\tau$ when it is strict and acts
bijectively on hom-sets, or when the $A_{\infty}$-category $\mathcal{A}$ is
directed and there is a bimodule map $\mathcal{A} (-, -) \to \mathcal{A} (-,
\tau (-))$ satisfying some hypotheses. Then we apply these results in order to
link together the Fukaya $A_{\infty}$-category of a family of exact
Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the
circular contactization.
