If I remember correctly, I read that given a presheaf $P:\mathcal{C}^{op} \to Set$, it is possible to describe it as a limit of representable presheaves. Could someone give a description of the construction together with a proof?



Presheaves as limits of representable functors?
You mean "colimit of representable presheaves", not limit. Any limits that C has are preserved by the Yoneda embedding. So if C is, say, a complete poset like • → •, so that it is small and has all limits, you won't be able to produce any non-representable presheaves by taking limits of representable ones.


The way to write any presheaf as a colimit of representables is, like all things Yoneda-related, somewhat tautological, and should be worked out for oneself; but anyways it's explained to some extent at [this nlab page](http://ncatlab.org/nlab/show/presheaf#properties_of_presheaves_4). Rather than write out formulas, I usually think of the example of simplicial sets: every simplicial set X can be formed as a colimit of its simplices, i.e., a diagram of representables which is indexed on the "category of simplices of X", whose objects are pairs (n, x) where n is in the indexing category and x is an object of Xn. The same works in any presheaf category.


