the upper adjoint. By the adjoint functor theorem, a monotone map between any pair of preorders preserves all joins if and only if it is a lower adjoint, and preserves all meets if and only if it is an upper adjoint. As such, each join-preserving morphism determines a unique upper adjoint in the inverse direction that preserves all meets. Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms. This also yields the insight that the introduced morphisms do basically describe just two different categories of complete lattices: one with complete homomorphisms and one with meet-preserving functions (upper adjoints), dual to the one with join-preserving mappings (lower adjoints). A particularly important special case is for lattices of subsets P(X) and P(Y) and a function from X to Y. In this case, the direct image and inverse image maps between the power sets are upper and lower adjoints to each other, respectively. Free construction and completion Free "complete semilattices" As usual, the construction of free objects depends on the chosen class of morphisms. Let us first consider functions that preserve all joins (i.e. lower adjoints of Galois connections), since this case is simpler than the situation for complete homomorphisms. Using the aforementioned terminology, this could be called a free complete join-semilattice. Using the standard definition from universal algebra, a free complete lattice over a generating set S is a complete lattice L together with a function i:S→L, such that any function f from S to the underlying set of some complete lattice M can be factored uniquely through a morphism f° from L to M. Stated differently, for every element s of S we find that f(s) = f°(i(s)) and that f° is the only morphism with this property. These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets. Free complete lattices in this sense can be constructed very easily: the complete lattice generated by some set S is just the powerset 2S, i.e. the set of all subsets of S, ordered by subset inclusion. The required unit i:S→2S maps any element s of S to the singleton set {s}. Given a mapping f as above, the function f°:2S→M is defined by .