I just wrote, much too quickly:
At the risk of being off the point, I think that the colimits that exist might not have been studied much because they are often not the 'right' ones, in a sense. As an example, the polynomial functor Set -> Set, X \mapsto X^2 (represented by 1 <- 2 -> 1 -> 1) has two automorphisms (the identity and the twist), and if I am not mistaken the identity functor X \mapsto X is the coequaliser of those two in the category of polynomial functors and their strong natural transformations (just because 1 is the equaliser of the two set auts 2 -> 2).
But the last sentence is of course pure nonsense. The equaliser of the two set auts 2 -> 2 is 0, and the conclusion is then that the constant polynomial functor X \mapsto 1 (represented by 1 <- 0 -> 1 -> 1) is the coequaliser. Sorry for the nonsense. I don't know where I had my head. I can hardly trust myself anymore, but if this second version is correct, it still illustrates the point I wanted to make, namely that the colimit is not the 'right' one.
The functor that 'ought' to be the coequaliser is of course X \mapsto X^2/2, which is not polynomial. (For example it does not preserve pullbacks.)
Cheers, Joachim. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]