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Every finitary endofunctor H of Set can be represented via a finitary signature and a collection of equations called "basic". We describe a terminal coalgebra for H as the terminal -coalgebra (of all -trees) modulo the congruence of applying the basic equations potentially infinitely often. As an application we describe a free iterative theory on H (in the sense of Calvin Elgot) as the theory of all rational -trees modulo the analogous congruence. This yields a number of new examples ofdoi:10.1016/j.ic.2005.11.005 fatcat:2otsod2ld5bwritbx7ly6h6kva