Hi Uwe, Not quite what you're asking for, but not too far either is the fact that for any monad T on ℂ, Kl(T) arises when factoring ℂ → T-Alg as identity-on-objects / fully faithful: ℂ → Kl(T) → T-alg. This is used in the literature on abstract nerve theorems, see, e.g., Familial 2-functors and parametric right adjoints by Mark Weber, or Polynomial functors and trees by Joachim Kock. This may be relevant to your ideas about presenting monads on graphs. The following papers might also be relevant. - Albert Burroni. Algèbres graphiques. Diagrammes, tome 7 (1982). - Dubuc and Kelly. A Presentation of Topoi as Algebraic Relative to Categories or Graphs. J. of Algebra 81 (1983). - Kelly and Power. Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. JPAA 89 (1993). Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]