<P> Certain classes of algebras enjoy both of these properties . The first property is more common, the case of having both is relatively rare . One class that does have both is that of multigraphs . Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges . The set H of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set . Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self - loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self - loop at one of the vertices . We can therefore organize the subgraphs of G as the multigraph Ω, called the power object of G . </P> <P> What is special about a multigraph as an algebra is that its operations are unary . A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s, t: E → V giving the source (start) and target (end) vertices of each edge . An algebra all of whose operations are unary is called a presheaf . Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets . Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier . Although the term "power object" is sometimes used synonymously with exponential object Y, in topos theory Y is required to be Ω . </P> <P> In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint . </P>

What is the power set of a null set