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) -> D.

For each object x from C, there is a functor

ev_x : Cat(C, D) -> D,

sending functor

F : C -> D

to F(x), and sending natural transformation

r : F -> G : C ->D

to 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.

--Tags:-[Mathematics]-----------