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A concrete description of terminal coalgebras, T , of all finitary endofunctors of Set is presented: each such a functor is a quotient of the polynomial endofunctor of some finitary signature Σ modulo some "basic" equations. Then T can be described as the algebra of all infinite Σ-labeled trees modulo the congruence obtained by applying the basic equations finitely or infinitely many times (a concept defined below). As a consequence, free iterative theories in the sense of Calvin Elgot aredoi:10.1016/s1571-0661(04)80629-7 fatcat:v7mzooorvffjtgcdiklbm6awue