We give a simple characterization of full subcategories of equational categories. If a \mathcal {a} is one such and B \mathcal {B} is the category of topological spaces, we consider a pair of adjoint functors a o p ⟷ U F B {\mathcal {a}^{op}}\underset {F}{\overset {U}{\longleftrightarrow }}\mathcal {B} which are represented by objects I and J in the sense that the underlying sets of U ( A ) U(A) and F ( B ) F(B) are a ( A , I ) \mathcal {a}(A,I) and B ( B , J ) \mathcal {B}(B,J) . (One may take I and J to have the same underlying set.) Such functors always establish a duality between Fix FU and Fix UF. We study conditions under which one can conclude that FU and UF are reflectors into Fix FU and Fix UF, that Fix FU = Image F = the limit closure of I in a \mathcal {a} and that Fix UF = Image U = the limit closure of J in B \mathcal {B} . For example, this happens if (1) a \mathcal {a} is a limit closed subcategory of an equational category, (2) J is compact Hausdorff and has a basis of open sets of the form { x ∈ J | α ( I ) ( x ) ≠ β ( I ) ( x ) } \{ x \in J|\alpha (I)(x) \ne \beta (I)(x)\} , where α \alpha and β \beta are unary a \mathcal {a} -operations, and (3) there are quaternary operations ξ \xi and η \eta such that, for all x ∈ J 4 , ξ ( I ) ( x ) = η ( I ) ( x ) x \in {J^4},\xi (I)(x) = \eta (I)(x) if and only if x 1 = x 2 {x_1} = {x_2} or x 3 = x 4 {x_3} = {x_4} . (The compactness of J may be dropped, but then one loses the conclusion that Fix FU is the limit closure of I.) We also obtain a quite different set of conditions, a crucial one being that J is compact and that every f in B ( J n , J ) \mathcal {B}({J^n},J) , n finite, can be uniformly approximated arbitrarily closely by a \mathcal {a} -operations on I. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed.