Inner properties of the category $JTF\sp{oo}$ of totally fuzzy presheaves when J is a general Heyting lattice are studied. It is proved that $JTF\sp{oo}$ has a final object, finite products, pullbacks, equalizers, and exponentials. The following negative result is also proved: if J is not antiordinal $JTF\sp{oo}$ has no monomorphism classifier.
Reviewer: V.V.Topencharov