In the community investigating coalgebras of functors F: K-> K (i.e., comma-categories Id/F ) the following condition in case K = Set is quite often used: Definition: A set functor is called bounded if there exists a cardinal k such that for every element a of every coalgebra there exists a subcoalgebra of cardinality less than k containing a . ("Bounded at k " is used for the given k.) So I wondered how this is related to accessibility, and here is the answer: Proposition: A set functor is bounded iff it is accessible. In fact, if k is an infinite, regular cardinal, then k-accessible = bounded at k . For, suppose that F is bounded at k , then we prove that in the usual category of elements of F for every object (A,a) there exists a morhpism m: (B,b) -> (A,a) with card B < k . It then follows easily that F is k-accessible (see e.g. 2.17 in the book of Rosicky and myself). We can assume A nonempty, and choose x in A. Let p:A -> FA be the coalgebra given by the constant map to a . There exists a subcoalgebra m: (B,q) -> (A,p) for some q: B -> FB with card B < k such that x = m(y) for an element y of B . Then b = q(y) clearly fulfills a = Fm(b), as requiered. Conversely, suppose that F is k-accessible. Given a coalgebra p: A -> FA , consider the collection m_i: A_i -> A of all subsets of cardinality less than k . Since F preserves k-directed unions, for each i there exists an i' such that m_i is contained in m_i' and the image of p.m_i is contained in that of Fm_i' . After iterating i -> i' -> (i')' ... k times we find a j such that m_i is contained in m_j , and the image of p.m_j is contained in that of Fm_j. The latter means that p can be restricted to p_j: A_j -> FA_j forming a subcoalgebra of the coalgebra above. Since every element of FA is contained in the image of some Fm_i , this proves that F is bounded at k . xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx