A functor $C \to D$ between categories induces a morphism of presheaf categories $Pre(D) \to Pre(C)$. This functor has a left adjoint given by left Kan extension and I am interested in knowing when this left adjoint preserves pull-back squares. 


I'm interested in any conditions that make this happen, but I am particularly interested in a special case. Let me say a little more about the context I am working in, and why I am interested. In the situation that this came up $C$ is a "lluf" subcategory of $D$, that is $C$ has the *same objects* as $D$ and the functor $C \to D$ is *faithful*. In that case it is good to call the functor $U:Pre(D) \to Pre(C)$ the forgetful functor. It automatically preserves limits and colimits. Let L be its left-adjoint.


Since C has the same objects as D, this forgetful functor is also *conservative*, meaning that it reflects isomorphisms. So by general non-sense (specifically [Beck's monadicity theorem](http://ncatlab.org/nlab/show/monadicity+theorem)) this is a monadic adjunction. This means that $Pre(D) = T-alg$ is the category of T-algebras in $Pre(C)$ where T is the monad $T= UL$. 


I'm trying to understand conditions under which this monad is *cartesian* in the sense [described at the n-lab](http://ncatlab.org/nlab/show/cartesian+monad). This means, among other things, that the monad T is supposed to send fiber products to fiber products. This is equivalent to having L send fiber products to fiber products. I want to understand when this happens. Does it always happen? Are there reasonable conditions on C or D that ensure that this happens?


Notice that I am not asking for L to be "left-exact", i.e. to commute with all finite limits. This property is generally much too strong. In particular L will not usually preserve terminal objects. This means it doesn't preserve products, but should instead send products to fiber products over $L(1)$. 


Here is an example. Let $C = pt$ be the singleton category and $D = G$ be the one object category with morphisms a group G. There is a unique inclusion $C \to D$ which is obviously faithful. The forgetful functor 
 $$U:Pre(D) \to Pre(C)$$ 
sends a G-set to its underlying set. The left adjoint L sends a set S to the free G-set $S \times G$. This doesn't preserve terminal objects, but it does commute with fiber products. What is more, the monad $T=UL$ is a classic example of a cartesian monad in the n-lab sense. 


I've played around with this, but can't seem to get it to work. I feel like this is going to be a well known result or there is going to be a counter example which sheds light on the situation. 



> 
> **Question**: In the context I described above (where $C \to D$ is lluf), does the left adjoint $$L: Pre(C) \to Pre(D)$$ always commute with fiber products? If not what is a counter example, and are there conditions one can place on C and D to ensure that L does commute with fiber products?
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An elementary question about adjunctions between presheaf categories preserving pullbacks. 
$\newcommand{\C}{\mathbf{C}} \newcommand{\D}{\mathbf{D}} \newcommand{\Lan}{\mathrm{Lan}} \newcommand{\yon}{\mathbf{y}} \newcommand{\CC}{[\C^\mathrm{op},\mathbf{Set}]} \newcommand{\DD}{[\D^\mathrm{op},\mathbf{Set}]}$ 
Expanding on my comment above:


Define: a category is *semi-filtered* iff every pair of arrows $x \leftarrow z \rightarrow y$ can be completed to a commutative square, and every parallel pair of arrows $f,g \colon x \to y$ have some $h : y \to w$ with $hf = hg$; equivalently, if every finite connected diagram has some co-cone under it. (Afaik, this isn't standard terminology; I don't recall seeing this property discussed, but it almost certainly has been.)


It's *filtered* if moreover it's non-empty and every pair of objects $x,y$ is connected by some $x \to w \rightarrow y$; equivalently, if every finite diagram has some co-cone under it.


**Answer**: for $f \colon \C \to \D$, the left Kan extension $f\_\* \colon \CC \to \DD$ will preserve pullbacks (equivalently, all connected finite limits) exactly if the opposite of its every comma category $(f \downarrow d)$ is semi-filtered. 


This is a close variant of the standard lemma (see e.g. Mac Lane and Moerdijk *Sheaves in Geometry and Logic*) that $f\_\*$ preserves pullbacks and the terminal object (equivalently, all finite limits) iff the opposite of every $(f \downarrow d)$ is filtered, i.e. if $f$ is *flat*.


**Proof**: the values of $f\_\*$ can be computed as colimits over the opposites of comma categories (see [this answer](https://mathoverflow.net/questions/32791/how-is-the-right-adjoint-f-to-the-inverse-image-functor-f-described-for-f/32808#32808)). But in $\mathbf{Set}$, finite limits commute with filtered colimits; and similarly, pullbacks commute with semi-filtered colimits.


(The first of these facts is standard. The second follows because in a semi-filtered category, each connected component is filtered; so a semi-filtered colimit is a coproduct of filtered colimits; and pullbacks commute with both coproducts and filtered colimits.)


