  Category theory reveals that many of these constructions are in fact certain objects in a category having a “universal property”. Indeed, from a categorical point of view, a Cartesian product in set theory, a direct product of groups (Abelian or otherwise), a product of topological spaces, and a conjunction of propositions in a deductive system are all instances of a categorical product characterized by a universal property. Formally, a product of two objects \(X\) and \(Y\) in a category \(\mathbf{C}\) is an object \(Z\) of \(\mathbf{C}\) together with two morphisms, called the projections, \(\boldsymbol{p}: Z \rightarrow X\) and  \(\boldsymbol{q}: Z \rightarrow Y\) such that—and this is the universal property—for all objects \(W\) with morphisms \(\boldsymbol{f}: W \rightarrow X\) and \(\boldsymbol{g}: W \rightarrow Y\), there is a unique morphism \(\boldsymbol{h}: W \rightarrow Z\) such that \(\boldsymbol{p} \circ \boldsymbol{h} = \boldsymbol{f}\) and \(\boldsymbol{q} \circ \boldsymbol{h} = \boldsymbol{g}.\)