A small example, but I think it's nice.  The generating function $C(t) = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} t^{2n}$ of the Catalan numbers is defined by the identity $C(t) = 1 + t^2 C(t)^2$.  So one might try to find a "Catalan object" in some category satisfying an isomorphism generalizing this identity.  One can take the corresponding combinatorial species in the sense of Joyal, but another choice is to take the <a href="http://qchu.wordpress.com/2010/04/17/graded-representation-theory/">invariant subalgebra of the tensor algebra of the defining representation of $\text{SU}(2)$</a>!