In these days I am studying some properties of Kan extensions. In order to do so I am looking at some general observations on Kan extensions that I would like to share with the community. A feedback would be very important to me.

Give a look at the following diagram to set the notation.

$\require{AMScd}$ \begin{CD} A @>f>> C\\ @VgVV \\ B \end{CD}

I will suppose that $g$ is fully faithful, that $C$ is cocomplete and that the kan extension is pointwise.

My interest is to understand how properties of $f$ interact with properties of Lan$_g f.$

To do so, I would start by giving a factorization of the Kan extension in the following way:

$$B \stackrel{\text{Elts}^g}{\to} \text{Fib}(A) \stackrel{f_*}{\to} \text{Fib}(C) \stackrel{\text{colim}}{\to} C.$$

Let's give a description of these functors.

Elts$^g$: $B \to \text{Fib}(A)$

By e $\text{Fib}(A)$ we mean the category of fibrations over $A$. This functor takes an object $b \in B$ into the category Elts($B(g\_, b)$).

The functor $f^*$ is just composition. So we map the fibration $E \to A$ to $E \to A \stackrel{f}{\to} C.$

Colim is just the functor that takes the colimit of the diagram induced by the fibration.

Under these notations it looks to me (please confirm, I am following the construction presented by Borceux (3.7.4, HoCA vol II)) that one can writhe down the equation: $$\text{Lan}_gf = \text{colim} \circ f_* \circ \text{Elts}^g. $$

Here come my questions:

What do we know about the functors

Elts$^g$ andcolim? More precisely,

Q1Are they faithful or conservative?

Q2What (co)limits do they preserve?

As a final remark, I would point out that any reference about Kan extensions which goes deeply in their properties and their behave is absolutely welcome.

**Motivations:**

My motivating questions are the following ones:

In the setting of the question, under what assumptions can I hope that $f$ conservative implies that Lan$_g f$ is conservative?

What about faithfulness?

Again, any reference for an answer to these questions is absolutely welcome.

These questions are not so desperate ad they may appear. If $B = \text{Set}^{A^{\text{op}}}$ and $g$ is the yoneda embedding, then the functor Elts$^y$ should be faithful and conservative because $$\text{PseudoPres}(A) \cong \text{Fib}(A) $$ precisely under that functor.