A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category contains, in particular, the small presheaves), but this is a stronger condition than small-cocompleteness in general.
It is natural to want to view totality as a cocompleteness condition. This is conceptually how totality is often introduced, and is justified to some extent by the following result of Kelly (Theorem 5.3).
Theorem (Kelly). The following are equivalent for a locally small category $A$.
- $A$ is total.
- For each presheaf $\phi \colon A^\circ \to \mathbf{Set}$, $A$ admits the weighted colimit $\phi * 1_A$.
- For each presheaf $\phi \colon K^\circ \to \mathbf{Set}$ and functor $d \colon K \to A$, $A$ admits the weighted colimit $\phi * d$ if and only if $\mathbf{Set}$ admits the weighted colimit $(\phi * d)(a, d{-})$ for each $a \in A$.
However, these conditions characterise total categories under a class of weights and diagrams; in other words, the colimits admitted by a total category $A$ are defined in terms of $A$. This is in contrast to free cocompletions, where we characterise colimits solely in terms of a class of weights.
My question. Is there a class $\Phi$ of weights such that the locally small $\Phi$-cocomplete categories are precisely the total categories?
Here, weight is simply defined to be a presheaf (if necessary, class-valued rather than set-valued), not necessarily with small domain. Such a $\Phi$, if it exists, will necessarily contain all small weights, and some large weights.
I would alternatively be happy with an explicit description of the universal property of the category of presheaves, in terms of a free cocompletion under small colimits together with some class of large colimits. In particular, I am predominantly wondering whether the presheaf construction on locally small categories forms a relative lax-idempotent pseudomonad.
(I suspect this question has a trivial answer, but for now I do not see whether it is trivially true or trivially false.)
