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.
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