In the title, $\operatorname{id}$ is a functor. Or $\operatorname{id}(C)$, to explicitely write a category $C$. So, a limit of all objects and morphisms in $C$. I am in doubt, is it proper to take such a big limit? Below I explain that $C$ in the motivating example is the category of algebras which usually is infinite. The question resides above, but if you can tell me something about the motivating example, you are welcome.
I found a proof that if such a limit exists, it is an initial object in $C$. The apex of the limit is an initial object and the morphisms of the limit are catamorphisms (this is just a special name for that unique morphism that goes from any initial object). This occurs in programming. $C$ is the category of algebras, the limit $L$ of $\operatorname{id}(C)$ is a set (?) where every element is a function that picks for every algebra an element of the carrier of that algebra. Intuitively, the function folds hidden something by a given algebra. Sometimes it's desirable to keep this function rather then an element of some initial algebra $0$. We go from any element $x$ of $0$ to the element of $L$ by feeding $x$ to morphisms of $L$. We go from any element $y$ of $L$ to the element of $0$ by picking the component of $y$ that corresponds to $0$.
