  We prove the compatibility of pushforward along a proper morphism of an
\'{e}tale constructible sheaf and the pushforward of its characteristic cycle
up to $p$-torsion. This was conjectured by Takeshi Saito. For this, we revisit
the construction of the characteristic cycle, due to Saito and Beilinson, from
more homotopical point of view. In particular, the language of
$\infty$-categories is indispensable to carry this out.
