Thm. 5.6 in Kelly gives a hierarchy of other properties related to "total" (at least in the case of locally essentially small categories):
total
implies
compact: Every functor $C^{op} \to Set$ which preserves all limits (even any large limits which exist) is representable
implies
hypercomplete: Every diagram $X : I \to C$ (with $I$ not necessarily essentially small) such that for every object $Y$, the collection of cones $Y \to X_i$ is bijective to a set, has a limit
implies
complete and mono-complete: The latter is the property that any ultra-wide pullback of monomorphisms has a limit (ultra-wide meaning the collection of monomorphisms does not necessarily have to be bijective to a set)
At some point, it could be interesting to add the other properties in this hierarchy (perhaps choosing a different name from "compact" since that seems likely to conflict). In fact, some of the proofs about to be added for failures to be cototal actually amount to proofs that the categories are not hypercomplete.
Thm. 5.6 in Kelly gives a hierarchy of other properties related to "total" (at least in the case of locally essentially small categories):
total$C^{op} \to Set$ which preserves all limits (even any large limits which exist) is representable$X : I \to C$ (with $I$ not necessarily essentially small) such that for every object $Y$ , the collection of cones $Y \to X_i$ is bijective to a set, has a limit
implies
compact: Every functor
implies
hypercomplete: Every diagram
implies
complete and mono-complete: The latter is the property that any ultra-wide pullback of monomorphisms has a limit (ultra-wide meaning the collection of monomorphisms does not necessarily have to be bijective to a set)
At some point, it could be interesting to add the other properties in this hierarchy (perhaps choosing a different name from "compact" since that seems likely to conflict). In fact, some of the proofs about to be added for failures to be cototal actually amount to proofs that the categories are not hypercomplete.