Limits of small functors

For a small category KK enriched over a suitable monoidal category VV, the free completion of KK under colimits is the presheaf category [Kop,V]. If KK is large, its free completion under colimits is the VV-category PKPK of small presheaves on KK, where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PKPK.