product and equalizer imply existence of a limit

Let be a category, a diagram and

1. the discrete subcategory of with the induced functor
   
2. the discrete category containing morphisms as objects for arbitrary
   
3. the functor with
   

Suppose that the categorical products for and equalizers exist
Then the limit for exists.

cumbersome notation :(

Let

then for with , we define by following universal property

Furthermore, for we define by following universal property

Then the equalizer

is the desired limit with

i.e.

(cf. morphism to product and each component)

• sloppy