### 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.