We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of familiar mathematical structures such as non-empty sets, groups, and topological spaces, as well as a method to construct an orthomodular poset from the complementary pairs of elements of a bounded modular lattice.