In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23):
Definition: A complete differential graded Lie algebra is an inverse limit of finite-dimensional, nilpotent differential graded Lie algebras (dgla).
The limit here is taken in the category of dglas, which is complete (and cocomplete). However, a couple of lines later, they show that complete dglas are pronilpotent assuming, apparently without loss of generality, that any complete dgla is a sequential limit of finite-dimensional nilpotent dglas, which I strongly believe is a more stringent condition that being an arbitrary limit. So what am I missing? Is the definition incorrect as stated (in the sense that we allow only some special kinds of limits), is the proof that complete dglas are pronilpotent wrong, or is there some implication or equivalence between various kinds of limits in the category of dglas that I don't know about?
In particular, I am interested to the model structure that they put on the category of complete dglas a bit later in the paper (page 36). We need the category to be complete and cocomplete, and if we can take arbitrary limits then completeness is obvious, while if we only allow e.g. sequential limits I'm not sure it holds.
I will be grateful for any contribution.
Edit: Cross-posted to MathOverflow.