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 invariant subalgebra of the tensor algebra of the defining representation of $\text{SU}(2)$!