\( \def\Lim{\operatorname{Lim}} \) \( \def\EV{\operatorname{EV}} \) \( \def\Cat{\operatorname{Cat}} \)
limit and evaluation

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.