The problem of “completing” a category is discussed in LNM0024. Here Lambek discusses notions like sup-complete and many other things, and also some drawbacks of the naive presheaves approach (some stuff is not preserved by this). A related problem was mentioned by Bondal in the Grothendieck conference talk Jan 2009, and he used this to motivate the introduction of Groth topologies, and the use of sheaves instead of presheaves.
nLab page on Completions of categories