  We ascertain conditions and structures on categories and semigroups which
admit the construction of pseudo-products and trace products respectively,
making their connection as precise as possible. This topic is modelled on the
ESN Theorem and its generalization to ample semigroups. Unlike some other
variants of ESN, it is self-dual (two-sided), and the condition of commuting
projections is relaxed. The condition that projections form a band (are closed
under multiplication) is shown to be a very natural one. One-sided reducts are
considered, and compared to (generalized) D-semigroups. Finally the special
case when the category is a groupoid is examined.
