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 $Mon(F)$.