  A compact T-algebra is an initial T-algebra whose inverse is a final
T-coalgebra. Functors with this property are said to be algebraically compact.
This is a very strong property used in programming semantics which allows one
to interpret recursive datatypes involving mixed-variance functors, such as
function space. The construction of compact algebras is usually done in
categories with a zero object where some form of a limit-colimit coincidence
exists. In this paper we consider a more abstract approach and show how one can
construct compact algebras in categories which have neither a zero object, nor
a (standard) limit-colimit coincidence by reflecting the compact algebras from
categories which have both. In doing so, we provide a constructive description
of a large class of algebraically compact functors (satisfying a
compositionality principle) and show our methods compare quite favorably to
other approaches from the literature.
