The eventual image of an endomorphism $f:A\to A$ is the intersection $\bigcap_{n=0}^{\infty} f^{(n)}(A)$ of the images of its iterates?. This makes sense in many different categories.
On the category of finite sets, the operation assigning to each endomorphism its eventual image is a dinatural transformation.
This example can also be viewed in terms of the trace of FinSet, defined as $\int^{a: FinSet}\; \hom(a, a)$. Indeed, the value of $f \in \hom(a, a)$ under the canonical map $\hom(a, a) \to \int^a \; \hom(a, a)$ is the same as that of the restriction $f|: Evim(f) \to Evim(f)$ to its eventual image, which is a permutation, and the value may be regarded as the conjugacy class of that permutation.
Tom Leinster, The eventual image (blog post)
Tom Leinster, The eventual image, part 2 (blog post)