Higher-order Algebras and Coalgebras from Parameterized Endofunctors

The study of algebras and coalgebras involve parametric description of a family of endofunctors. Such descriptions can often be packaged as parameterized endofunctors. A parameterized endofunctor generates a higher-order endofunctor on a functor category. We characterize initial algebras and final coalgebras for these higher-order endofunctors, generalizing several results in the literature.