  The Dowker theorem is a classical result in the topology of finite spaces,
claiming that any binary relation between two finite spaces defines two
homotopy-equivalent complexes (the Dowker complexes). Recently, Barmak
strengthened this to a simple-homotopy-equivalence. We reprove Barmak's result
using a combinatorial argument that constructs an explicit acyclic matching in
the sense of discrete Morse theory.
