Let \(D\) be a category with functorial limits of functors from some category \(C\). That is, there is a functor
$$\lim : \Cat(C, D) \to D.$$For each object \(x\) from \(C\), there is a functor
$$ev_x : \Cat(C, D) \to D,$$sending a functor \(F : C \to D\) to \(F(x)\), and sending a natural transformation \(r : F \to G : C \to D\) to its component at \(x\) (\(r_x : F(x) \to G(x)\)). By mapping \(x\) to \(ev_x\) and morphism \(f : x \to y\) to an obvious natural transformation from \(ev_x\) to \(ev_y\) we obtain a functor \(\EV : C \to \Cat(\Cat(C, D), D)\).
Alternatively, any meaningful generic diagram commutes.
Amusing and trivially checkable fact is that \(\lim = \Lim \EV\). That is, limits are always point-wise.