Dear all, many thanks for the very useful replies concerning my question about Grothendieck-Yoneda-Colimits. Now another question on top of it: I'm more on the "applied side" and interested in syntactic representation of things. For a many-sorted algebraic signature \Sigma with a finite set (discrete category) S of sorts the construction of \Sigma-terms gives us a monad T_\Sigma:Set^S -> Set^S. The syntactic category with S^* as set of objects, finite tuples of terms as morphisms and "composition by substitution" (Lawvere) can be seen as a subcategory of the Kleisli category of this monad. We generalized recently the concept of algebraic signatures and algebras to graphs: input and out put arities of operations are graphs as well as the carriers of algebras are graphs. We describe the construction of "graph terms" and get a monad on Set^B with B the category given by two parallel arrows s,t:E->V. What we would like to have is a nice generalization of the construction of syntactic Lawvere categories to this case. I learned now that "the category [C^op,Set] is the free colimit completion of C". My question is, if there are similar results for the Kleisli category of a monad on [C^op,Set]? Best regards Uwe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]