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