# Terminology question: "core limit"

I am looking for the correct terminology and a reference to the following construction.

Let $$F:\mathcal C\to\mathcal D$$ be a functor. Consider "cones" $$(G,\alpha)$$ where $$G:\mathcal C\to\mathcal D$$ is a functor such that all morphisms $$G(f)$$, $$f\in Mor(\mathcal C)$$, are invertible (i.e. $$G$$ is a functor to the core of $$\mathcal D$$), and $$\alpha: G\Rightarrow F$$ is a natural transformation. The "core limit" of $$F$$ is the universal "cone" $$G_u$$.

The "core limit" $$G_u$$ defines a "core limit object" $$\hbox{core-lim } F=G_u(c)$$ and a "monodromy group" $$Mon(F)=G_u(\hom(c,c))$$, $$c\in\mathcal C$$, which are unique up to isomorphism when $$\mathcal C$$ is connected. In this case, the real limit $$\lim F$$ is the invariant of $$\hbox{core-lim } F$$ by the action of $$Mon(F)$$.

I don't know if there is a standard terminology for this specific construction, but let me nonetheless say something. Let $$|C|$$ denote the groupoidification of $$C$$, i.e. the initial groupoid with a map $$p:C\to |C|$$.
Then what you are describing is the right Kan extension of $$F: C\to D$$ along $$p: C\to |C|$$. This recovers what you said at the end about the invariants of $$G$$ being exactly the limit of $$F$$, because right Kan extensions compose, and $$\lim_C F$$ is the right Kan extension of $$F$$ along $$C\to *$$, which you can right Kan extend in two steps: first to $$|C|$$, and second from $$|C|$$ to $$*$$ - this second step takes invariants for $$G$$ when $$|C| = BG$$ for some group $$G$$, i.e. when $$C$$ is (weakly) connected, but in general takes $$\prod_{x\in\pi_0|C|} (\mathrm{core-lim})^{Aut(x)}$$.