We consider the notion of `universal-algebraic' functor. These are functors on many-sorted varieties that preserve sifted colimits. Universal algebra stems from the doctrine of  finite products, and sifted colimits are precisely those colimits that commute in $\Set$ with finite products. Universal-algebraic functors on many-sorted varieties can be presented by operations and equations. We exploit the flexibility provided by this notion and discuss several applications. 
\par First, it can be used to study universal algebra over the category $\Nom$ of nominal sets.  The signatures for algebras over $\Nom$ are given by functors having presentations by operations and equations.  We prove an HSP-like theorem for algebras over nominal sets, using completely standard universal algebra and the fact that $\Nom$ is a full reflective subcategory of a many-sorted variety. Since our notion of signature is quite general, the equational logic obtained in our setting is more expressive than the nominal algebra logic of Gabbay and Mathijssen, or the nominal equational logic of Clouston and Pitts. However we can isolate a `uniform' fragment of our logic that corresponds to these logics, thus we give a new way of comparing the two different approaches.  
\par Second, we give a categorical approach to algebraic semantics of first-order logic. We show how to obtain algebraic models of first-order logic as
algebras for a functor on a many-sorted variety $\BA^{\fbb_+}$ satisfying some additional equations. These algebras are equivalent to the polyadic algebras of Halmos, and we can dualise them to obtain a coalgebraic semantics of first-order logic.
\par Third, this notion links the uniform treatment of logics for
coalgebras of an arbitrary type T with concrete syntax and proof systems. Analysing
the many-sorted case is essential for achieving modular completeness proofs for coalgebraic logics.