nLab
total category

A locally small category $C$ is total if its Yoneda embedding $C\to [C^{\mathrm{op}},\mathrm{Set}]$ has a left adjoint. If $C^{\mathrm{op}}$ is total, $C$ is called cototal.

The definition above requires some set-theoretic assumption to ensure that the functor category $[C^{\mathrm{op}},\mathrm{Set}]$ exists, but it can be rephrased to say that the colimit of ${\mathrm{Id}}_C:C\to C$ weighted by $W$ exists, for any $W:C^{\mathrm{op}}\to \mathrm{Set}$. (This still involves quantification over large objects, however, so some foundational care is needed.) This version has an evident generalization to enriched categories.

Total categories satisfy a very satisfactory adjoint functor theorem: any colimit-preserving functor from a total category to a locally small category has a right adjoint.

Any category which is monadic over Set is total, as is any category admitting a topological functor to Set.

References

• G. M. Kelly, A survey of totality for enriched and ordinary categories.