We explain how Iyama—Yang’s silting reduction is compatible with Buan—Marsh’s reduction of τ-rigid pairs. Then, we reconstruct the τ-cluster morphism category of a finite-dimensional algebra, or more generally of a non-positive proper differential graded algebra. Approaching τ-cluster morphism categories in terms of silting theory, as opposed to τ-tilting theory, has the advantage that the associativity of composition is proved more neatly. Time permitting, we explore the cubical structure of τ-cluster morphism categories and discuss alternative ways of defining them.