Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbf C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\mathbb C)$. These constructions factor through two others:

1. The free bicompletion $\mathbf b : \mathbb C \to \mathcal B(\mathbb C)$.
2. The Isbell envelope $\mathbf i :\mathbb C \to \mathcal I(\mathbb C)$.

Both constructions have universal properties: the former freely adds limits and colimits; whilst the latter freely constructs a cylinder factorisation system.

However, it is not clear to me exactly how the two relate. May one construction be expressed in terms of the other (e.g. is there a factorisation of free bicompletion through the Isbell envelope)? If not, is there some other relationship?

Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\mathbb C)$. These constructions factor through two others:

1. The free bicompletion $\mathbf b : \mathbb C \to \mathcal B(\mathbb C)$.
2. The Isbell envelope $\mathbf i :\mathbb C \to \mathcal I(\mathbb C)$.

Both constructions have universal properties: the former freely adds limits and colimits; whilst the latter freely constructs a cylinder factorisation system.

However, it is not clear to me exactly how the two relate. May one construction be expressed in terms of the other (e.g. is there a factorisation of free bicompletion through the Isbell envelope)? If not, is there some other relationship?