Let 𝒞 be a cocomplete category and let *U* : ℐ → 𝒞 be a small diagram. Then there is an induced realisation–nerve adjunction,

*R* ⊣ *N* : 𝒞 → [ℐ^{op}, **Set**]

where *N*(*C*) is the presheaf 𝒞(*U*–, *C*) and *R* is defined by left Kan extension. The comonad induced by the realisation–nerve adjunction is precisely the density comonad induced by *U*, so it is natural to ask, when is *R* comonadic?

Beck’s monadicity theorem implies *R* is comonadic if and only if *R* is conservative and preserves certain equaliser diagrams in [ℐ^{op}, **Set**], and Thomas Athorne described some sufficient conditions in his thesis. We discuss here his condition for conservativity.

Let us say that *U* satisfies the **repeated element condition** if the components of the adjunction unit are monomorphisms in [ℐ^{op}, **Set**]. Unravelling the definitions, this amounts to saying *R* reflects equality between presheaf morphisms with representable domain.

First, suppose *R* is conservative. Since [ℐ^{op}, **Set**] has coequalisers, *R* must also be faithful. Thus, *R* reflects monomorphisms, and in particular, *U* satisfies the repeated element condition.

Conversely, suppose *U* satisfies the repeated element condition. By general nonsense about adjunctions, this happens if and only if *R* is faithful. Since [ℐ^{op}, **Set**] is a balanced category, to verify that *R* reflects isomorphisms, it is enough to show that *R* reflects epimorphisms. This is easily done by considering cokernel pairs.

We may therefore conclude that the following are equivalent:

*U* satisfies the repeated element condition.
*R* reflects monomorphisms.
*R* is faithful.
*R* is conservative.