# On coslices and commas of locally finitely presentable categories

Axel Osmond

(March 2021)

###### Abstract

We give an explicit description of the generator of finitely presented objects
of the coslice of a locally finitely presentable category under a given
object, as consisting of all pushouts of finitely presented maps under this
object. Then we prove that the comma category under the direct image part of a
morphism of locally finitely presentable category is still locally finitely
presentable, and we give again an explicit description of its generator of
finitely presented objects. We finally deduce that 2-category ${\bf LFP}$ has
comma objects computed in ${\bf Cat}$.

## Introduction

In this note we investigate in detail the comma and coslice construction in
the context of locally finitely presentable categories.

It is well known since [1] that coslices of locally finitely presentable
categories are again locally finitely presentable categories; however the
proof involves an abstract characterization from which one cannot retrieve an
explicit generator of finitely presented objects. However, in the context of
the _spectral construction_ arose a similar problem of characterising finitely
presentable left maps under a given object - which are used to construct the
_spectral site_ associated to this object - and this led to ask in particular
what could be a good choice of generator in a coslice. After discussing this
topics at several occasion with members of the category theory community, it
appeared that we were still lacking a reference containing the explicit
result, though elements of answer were already present in two previous works,
[2] and [5], and this kind of result might have been folklore for a few
connoisseurs. We though it could be useful to provide an answer as detailed
and concrete as possible to fix this lack of reference, which is the purpose
of first half of this work.

The second half of this paper is devoted to the construction of comma
categories in the 2-category ${\bf LFP}$ of locally finitely presentable
categories. Concretely, we prove that if $F:\mathcal{A}\rightarrow\mathcal{B}$
is a morphism of locally finitely presentable categories, then the comma
$F_{*}\downarrow\mathcal{B}$ is locally finitely presentable and define
together with its canonical projections a comma object in ${\bf LFP}$. It
seems that until now the only elements of answer on this topic was
[7][Proposition 6.1.1] and a subsequent adaptation in [1], which however was a
weaker statement only proving accessibility of comma object without any
control of the rank of accessibility. We improve this result by proving that
in the situation above the comma construction needs not raising the rank of
accessibility, thanks to a characterization of finitely presented objects of
the comma. Moreover, we prove that this comma object calculated in ${\bf Cat}$
actually inherits the universal property of the comma object in ${\bf LFP}$.

Generalities about coslices and few technicalities are gathered in the first
part of this work.

## Acknowledgements

The author would like to express special thanks to Mathieu Anel for his help
on this topic and several important discussions, and also useful remarks on a
previous draft of this work; he should also emphasize the key influence of [2]
on this work. He is also specially grateful to Ivan Di Liberti for several
helpful and stimulating discussions on this problem. He also thanks Paul-André
Melliès, Clemens Berger and Jiri Adamek for past discussions on this topic.

##  1 Generalities about coslices and comma categories

In this section we first recall very basic facts about coslice categories.

######  Definition 1.1.

Let be $\mathcal{A}$ a category; the _coslice_ of $\mathcal{A}$ at an object
$A$ is the category $A\downarrow\mathcal{A}$ whose

  * $-$

objects are arrows $f:A\rightarrow B$ in $\mathcal{A}$

  * $-$

morphisms $f_{1}\rightarrow f_{2}$ are all arrows $g$ inscribed in a triangle
as below

${A}$${{B_{1}}}$${{B_{2}}}$$\scriptstyle{f_{1}}$$\scriptstyle{g}$$\scriptstyle{f_{2}}$

We denote as ${\rm cod}:A\downarrow\mathcal{A}\rightarrow\mathcal{A}$ the
_codomain functor_ sending an arrow $f:A\rightarrow B$ on its codomain $B$ and
a morphism $g:f_{1}\rightarrow f_{2}$ on the underlying arrow
$g:B_{1}\rightarrow B_{2}$.

Coslices are related to the following notion:

######  Definition 1.2.

For a category $\mathcal{A}$, the _arrow category_ $\mathcal{A}^{2}$ is the
category whose

  * $-$

objects are arrow $f:A\rightarrow B$

  * $-$

morphisms $f_{1}\rightarrow f_{2}$ are pairs $(g,g^{\prime})$ inscribed in a
square as below

${{A_{1}}}$${{A_{2}}}$${{B_{1}}}$${{B_{2}}}$$\scriptstyle{f_{1}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f_{2}}$$\scriptstyle{g}$

Similarly, the arrow category is equipped with a codomain functor ${\rm
cod}:\mathcal{A}^{2}\rightarrow\mathcal{A}$; it is moreover equipped with a
further _domain functor_ ${\rm dom}:\mathcal{A}^{2}\rightarrow\mathcal{A}$
sending $f:A\rightarrow B$ on $A$ and $(g,g^{\prime}):f_{1}\rightarrow f_{2}$
on the underlying arrow between domains $g:A_{1}\rightarrow A_{2}$.

Arrows categories inherit generally lot of the structure of the underlying
category. In particular, $\mathcal{A}^{2}$ inherits limits and colimits
existing in $\mathcal{A}$, and moreover they are _pointwise_ , that is, for
$F:I\rightarrow\mathcal{A}^{2}$, the limit cone (resp. the colimiting cocone)
in $\mathcal{B}^{2}$ are computed as

${{\underset{i\in I}{\lim}\,{\rm dom}\,F(i)}}$${{\underset{i\in I}{\lim}\,{\rm
cod}\,F(i)}}$${{{\rm dom}\,F(i)}}$${{{\rm
cod}\,F(i)}}$$\scriptstyle{p_{i}}$$\scriptstyle{F(i)}$$\scriptstyle{p_{i}^{\prime}}$$\scriptstyle{\underset{i\in
I}{\lim}\,F(i)}$ resp. ${{{\rm dom}\,F(i)}}$${{{\rm
cod}\,F(i)}}$${{\underset{i\in I}{{\textup{colim}}}\,{\rm
dom}\,F(i)}}$${{\underset{i\in I}{{\textup{colim}}}\,{\rm
cod}\,F(i)}}$$\scriptstyle{q_{i}}$$\scriptstyle{F(i)}$$\scriptstyle{q_{i}^{\prime}}$$\scriptstyle{\underset{i\in
I}{{\textup{colim}}}\,F(i)}$

where the projections $p_{i},p_{i}^{\prime}$ (resp. the inclusion
$q_{i},q_{i}^{\prime}$) form the limiting cone (resp. the colimiting cocone)
of the domains and codomains respectively, while the middle arrow is induced
by the universal property of the limit (resp. the colimit) from the induced
cone $(F(i)p_{i}:\lim_{i\in I}{\rm dom}\,F(i)\rightarrow{\rm cod}\,F(i))_{i\in
I}$ (resp. from the induced cocone $(q^{\prime}_{i}F(i):{\rm
dom}\,F(i)\rightarrow{\textup{colim}}_{i\in I}{\rm cod}\,F(i))_{i\in I}$). As
a consequence, both the functors ${\rm dom}$ and ${\rm cod}$ preserve either
limits and colimits.

We also have a common section $1:\mathcal{A}\rightarrow\mathcal{A}^{2}$,
sending an object $A$ on its identity arrow $1_{A}:A\rightarrow A$, endowed
with a string of adjunctions

${{\mathcal{A}^{2}}}$${{\mathcal{A}}}$$\scriptstyle{1}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{{\rm dom}}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

It is easy to see that $1$ also preserves both limits and colimits.

We recall that limits in the coslice $A\downarrow\mathcal{A}$ are computed
from the universal property of the limit: for a diagram $F:I\rightarrow
A\downarrow\mathcal{A}$ the limit of $F$ is computed as

${{\lim{\rm cod}\,F}}$${A}$${{{\rm
cod}\,F(i)}}$$\scriptstyle{F_{i}}$$\scriptstyle{p_{i}}$$\scriptstyle{(F(i))_{i\in
I}}$

In particular the codomain functor preserves limits.

Let us now turn our attention to the colimits in the coslices. For any object
$A$ in a cocomplete category $\mathcal{A}$, $A\downarrow\mathcal{A}$ has
colimits and they are computed as follows. Let $I\rightarrow
A\downarrow\mathcal{A}$ a functor; from the category $\overline{I}$ by adding
to $I$ a cone made of an object $i_{0}$ and for each $i\in I$ an arrow
$\overline{f}_{i}:i_{0}\rightarrow i$ such that for each $d:i\rightarrow j$ in
$I$ one has $d\overline{f}_{i}=\overline{f}_{j}$, and denote
$\iota:I\hookrightarrow\overline{I}$ the full inclusion. Then one can extend
canonically ${\rm cod}\,F$ into $\overline{F}$

${I}$${A\downarrow\mathcal{A}}$${\overline{I}}$${\mathcal{A}}$$\scriptstyle{F}$$\scriptstyle{\iota}$${=}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{\overline{F}}$

by defining $\overline{F}(i)={\rm cod}(F(i))$, $\overline{F}(i_{0})=A$ and
$\overline{F}(\overline{f}_{i})=F(i)$. Then $\overline{F}$ form a cone in
$\mathcal{A}$ with vertex $A$; now one has

${\rm cod}({\textup{colim}}\,F)\simeq{\textup{colim}}\,\overline{F}$

and the colimit in $A\downarrow\mathcal{A}$ is

${\textup{colim}}\,F=\overline{q}_{i_{0}}:A\rightarrow{\textup{colim}}\,\overline{F}$

which coincides also with any composite $\overline{q}_{i}F(i)$. So that
actually ${\rm cod}$ does not exactly preserve colimits, though any colimit in
$A\downarrow\mathcal{A}$ is sent to a colimit over a diagram canonically
extending it: in some sense, we could say that ${\rm cod}$ “corrects
colimits”. In particular we have a canonical arrow

${\textup{colim}}\;{\rm cod}\,F\rightarrow{\rm cod}({\textup{colim}}\,F)$

which may not be an isomorphism.

######  Remark.

Let us see concretely why the codomain functor ${\rm
cod}:A\downarrow\mathcal{A}$ does not preserve arbitrary colimits, failing in
particular to preserve binary coproducts. Suppose that $f_{1}:A\rightarrow{\rm
cod}(f_{1}),f_{2}:A\rightarrow{\rm cod}(f_{2})$ have coproduct
$f_{1}+f_{2}:A\rightarrow{\rm cod}(f_{1}+f_{2})$; then it may happen that, in
$\mathcal{A}$, the inclusions of the coproduct $\iota_{1}:{\rm
cod}(f_{1})\rightarrow{\rm cod}(f_{1})+{\rm cod}(f_{2}),\,\iota_{2}:{\rm
cod}(f_{2})\rightarrow{\rm cod}(f_{1})+{\rm cod}(f_{2})$ are such that the
composite $\iota_{1}f_{1}$ and $\iota_{2}f_{2}$ are not equal, while the
inclusion of the coproduct $q_{1}:f_{1}\rightarrow
f_{1}+f_{2},\,q_{2}:f_{2}\rightarrow f_{1}+f_{2}$ are such that
$q_{1}f_{1}=f_{1}+f_{2}=q_{2}f_{2}$ in $\mathcal{A}$: that is we have a
diagram

${{\rm cod}(f_{1})}$${A}$${{\rm cod}(f_{1})+{\rm cod}(f_{2})}$${{\rm
cod}(f_{1}+f_{2})}$${{\rm
cod}(f_{2})}$$\scriptstyle{\iota_{1}}$$\scriptstyle{q_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$${\neq}$$\scriptstyle{\langle
q_{1},q_{2}\rangle}$$\scriptstyle{\iota_{2}}$$\scriptstyle{q_{2}}$

where the outer square commutes as well as the upper and lower triangles, but
not the inner square. Then ${\rm cod}(f_{1}+f_{2})$ cannot be the coproduct
${\rm cod}(f_{1})+{\rm cod}(f_{2})$: in fact, it is the coequalizer ${\rm
cod}(f_{1}+f_{2})={\textup{coeq}}(\iota_{1}f_{1},\iota_{2}f_{2})$ in
$\mathcal{A}$. More generally, for an arbitrary diagram $f_{(-)}:I\rightarrow
A\downarrow\mathcal{A}$, we have that ${\rm cod}({\textup{colim}}_{i\in
I}f_{i})$ is the wide coequalizer of the set of parallel arrows
$(\iota_{i}f_{i}:A\rightarrow{\textup{colim}}_{i\in I}{\rm cod}(f_{i})$.

However for certain shapes of colimits, we have actually preservation by the
codomain functor. Before that, recall first that a category $I$ is said to be
_connected_ if for any $i,i^{\prime}$ there exists a finite zigzag or arrows
in $I$

${{i_{1}}}$${{i_{n}}}$${i}$${{...}}$${{i^{\prime}}}$

Now a functor $G:I\rightarrow J$ is said to be _final_ if for any $j$ in $J$
the comma category $i\downarrow G$ is non empty and connected. In particular,
whenever $I$ is filtered, this amounts to saying that for any $j$ in $J$ there
is some $d:j\rightarrow F(i)$ and that any span $d_{1}:j\rightarrow F(i_{1})$,
$d_{2}:j\rightarrow F(i_{2})$ can be completed by a commutative square

${j}$${{F(i_{2})}}$${{F(i_{1})}}$${{F(i_{3})}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{F(e_{2})}$$\scriptstyle{F(e_{1})}$

It is well known that a functor $G:I\rightarrow J$ is final if and only if
precomposing with it does not modify colimits: that is, if for any
$F:J\rightarrow\mathcal{A}$, we have an isomorphism

$\underset{J}{{\textup{colim}}}\,F\simeq\underset{I}{{\textup{colim}}}\,FG$

Equipped with those notions, we can say more on the codomain functor.

######  Lemma 1.3.

The codomain functor preserves connected colimits.

###### Proof.

This is because in this case the inclusion
$\iota:I\hookrightarrow\overline{I}$ is cofinal: indeed, for any $i,j$, the
unique span $\overline{f}_{i}:i_{0}\rightarrow i$,
$\overline{f}_{j}:i_{0}\rightarrow j$ can be completed by a zigzag because $I$
is connected, and we imposed that the $\overline{f}_{i}$ form a cone, so that
triangles in this zigzag commute. Hence we have

$\displaystyle{\rm cod}({\textup{colim}}\,F)$
$\displaystyle={\textup{colim}}\,\overline{F}$
$\displaystyle={\textup{colim}}\,\overline{F}\,\iota$
$\displaystyle={\textup{colim}}\,{\rm cod}\,F$

∎

######  Corollary 1.4.

If $\mathcal{A}$ has filtered colimits, then so has $A\downarrow\mathcal{A}$
and the codomain functor preserves them.

######  Corollary 1.5.

If $\mathcal{A}$ has coequalizers, then so has $A\downarrow\mathcal{A}$ and
the codomain functor preserves them.

######  Remark.

Thoses facts are well known (see for instance [3][Proposition 2.16.3]). For
coequalizers, observe that in the diagram below with the coequalizer computed
in $\mathcal{A}$

${{B_{1}}}$${{B_{2}}}$${{{\textup{coeq}}(g,g^{\prime})}}$${A}$$\scriptstyle{q_{g,g^{\prime}}}$$\scriptstyle{g}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$

the composite
$q_{g,g^{\prime}}f_{2}:A\rightarrow{\textup{coeq}}(g,g^{\prime})$ together
with $q_{g,g^{\prime}}$ is a coequalizer in $A\downarrow\mathcal{B}$.

To conclude this section, we should give a lemma that will be of use in the
following section:

######  Lemma 1.6.

Let be $F:\mathcal{I}\rightarrow\mathcal{J}$ a functor such that

  * $-$

$F$ is essentially surjective and full

  * $-$

$I$ is filtered

Then $F$ is cofinal and moreover $J$ is also filtered.

###### Proof.

For any $j$ in $J$, the comma $j\downarrow F$ is non empty, as it contains an
isomorphism $j\simeq F(i)$ given by essential surjectivity. Now, for any span

${j}$${{F(i_{2})}}$${{F(i_{1})}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$

then filteredness of $I$ ensures the existence of a cospan
$d_{1}:i_{1}\rightarrow i_{3}$, $d_{2}:i_{2}\rightarrow i_{3}$; however,
$F(d_{1})f_{1}$ and $F(i_{2})f_{2}$ may not be equal: however, essential
surjectivity gives some $u:j\simeq F(i)$ and fullness gives some
$e_{1}:i\rightarrow i_{1}$, $e_{2}:i\rightarrow i_{2}$ such that
$f_{1}=F(e_{1})w$ and $f_{2}=F(e_{2})w$, and $d_{1}e_{1}$ and $d_{2}e_{2}$ are
equalized in $I$ by some $d_{3}:i_{3}\rightarrow i_{4}$ so that the following
diagram commutes

${j}$${{F(i)}}$${{F(i_{2})}}$${{F(i_{1})}}$${{F(i_{4})}}$$\scriptstyle{F(d_{3}d_{1})}$$\scriptstyle{F(d_{3}d_{2})}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{w\atop\simeq}$$\scriptstyle{F(e_{2})}$$\scriptstyle{F(e_{1})}$

Hence $i\downarrow F$ is non empty and connected: $F$ is cofinal. Filteredness
of $J$ is inherited from $I$. ∎

##  2 Coslices of Locally finitely presentable categories

In this section we turn to coslices of locally finitely presentable
categories. It is already well known that coslices, as well as slices, of
locally finitely presentable categories are still locally finitely presentable
(and also more generally, (co)slices of accessible categories are accessible),
as it was for instance proven in [1][Proposition 1.57]. However this instance
seems to be the sole reference about it in the literature, though it processes
by a specific characterization of locally finitely presentable categories that
dispenses to care about explicit description of finitely presented object. It
seems that no other reference in the literature treated this topic nor offered
concrete proof, nor any explicit description of the finitely presented
objects. However this problem is related to a specific version of the small
object argument, as done in [2] and [5], which examine a generator of finitely
presented left map in the context of a factorization system. Inspired from
those sources, we provide here a characterization of the generator of finitely
presented objects in the coslices of a locally finitely presented category.

Before going any further, we recall the following generalities about locally
finitely presentable categories. We recall that a finitely presented object in
a category $\mathcal{B}$ is some $K$ such that for any
$F:I\rightarrow\mathcal{B}$ with $I$ filtered we have an isomorphism

$\mathcal{B}[K,\underset{i\in I}{{\textup{colim}}}\,F(i)]\simeq\underset{i\in
I}{{\textup{colim}}}\,\mathcal{B}[K,F(i)]$

Concretely this means that:

  * $-$

for any arrow $a:K\rightarrow{\textup{colim}}_{i\in I}\,F(i)$, there is some
$i\in I$ and some factorization, called a _lift_

${{F(i)}}$${K}$${{\underset{i\in
I}{{\textup{colim}}}\,F(i)}}$$\scriptstyle{q_{i}}$$\scriptstyle{a}$$\scriptstyle{b}$

(we shall often denote such a lift as the pair $(i,b)$);

  * $-$

and for any two such lifts $(b_{1},i_{1})$ and $(b_{2},i_{2})$ of a same arrow
$a$, we shall refer to as _parallel lifts_ , there exists a cospan
$d_{1}:i_{1}\rightarrow i$ and $d_{2}:i_{2}\rightarrow i$ such that both lifts
are equalized jointly into a third lift, we shall often call a _further
refinement_

${K}$${{F(i_{1})}}$${{F(i_{2})}}$${{F(i)}}$${{\underset{i\in
I}{{\textup{colim}}}\,F(i)}}$$\scriptstyle{F(d_{1})}$$\scriptstyle{F(d_{2})}$$\scriptstyle{q_{i}}$$\scriptstyle{q_{i_{2}}}$$\scriptstyle{q_{i_{1}}}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$

First the following lemma from [1][Exercice 1.o] says that finitely presented
objects can be jointly used to test if a cocone is colimitting:

######  Lemma 2.1.

Let be $D:I\rightarrow\mathcal{B}$ with $I$ finitary and $\mathcal{B}$ locally
finitely presentable. Then a cocone $(f_{i}:D_{i}\rightarrow C)_{i\in I}$ in
$\mathcal{B}$ exhibits $C$ as the colimit ${\textup{colim}}\,D$ if and only if
for any finitely presented object $K$ one has an isomorphism natural in $K$

$\mathcal{B}(K,C)\simeq\underset{i\in
I}{{\textup{colim}}}\;\mathcal{B}(K,D_{i})$

###### Proof.

This is because finitely presented objects from altogether a separating
family. If $C$ is such that we have in each $K$ the natural isomorphism above
then one has also a natural isomorphism

$\mathcal{B}(K,C)\simeq\mathcal{B}(K,\underset{i\in
I}{{\textup{colim}}}\;D_{i})$

Hence by naturallity we have an equivalence $\mathcal{B}_{\omega}\downarrow
C\simeq\mathcal{B}_{\omega}\downarrow{\textup{colim}}_{i\in I}D_{i}$, and
hence an isomorphism $C\simeq{\textup{colim}}_{i\in I}D_{i}$. ∎

We also will need the following, when encountering finitary functors, to
control the behaviour of their eventual left adjoints:

######  Lemma 2.2.

Let $F\dashv G$ be adjoint functors between locally finitely presentable
categories. Then $F$ maps finitely presented objects on finitely presented
objects if and only if $G$ is finitary.

###### Proof.

If $F:\mathcal{B}\rightarrow\mathcal{A}$ preserves finitely presented objects
and $D:I\rightarrow\mathcal{A}$ a filtered diagram, then for any finitely
presented object $K$ in $\mathcal{B}$:

$\displaystyle\begin{split}\mathcal{B}(K,G(\underset{i\in
I}{{\textup{colim}}}\;D_{i}))&\simeq\mathcal{A}(F(K),\underset{i\in
I}{{\textup{colim}}}\;D_{i})\\\ &\simeq\underset{i\in
I}{{\textup{colim}}}\;\mathcal{A}(F(K),D_{i})\\\ &\simeq\underset{i\in
I}{{\textup{colim}}}\;\mathcal{B}(K,G(D_{i}))\end{split}$

But from lemma 2.1 this means that $G({\textup{colim}}_{i\in
I}D_{i})\simeq{\textup{colim}}_{i\in I}G(D_{i})$. Then converse is immediate:
if $G$ is finitary, $K$ is finitely presented and $D:I\rightarrow\mathcal{A}$
is filtered, then permuting the identities above ensures that $F(K)$ still is
finitely presented. ∎

Then we should give some precision on the arrow category of a locally finitely
presentable category, and more generally, about finitely presented object

######  Proposition 2.3.

If $\mathcal{B}$ is locally finitely presentable, then for any finite diagram
$D$, the functor category $\mathcal{B}^{D}$ is locally finitely presentable
and we have

$(\mathcal{B}^{D})_{\omega}\simeq(\mathcal{B}_{\omega})^{D}$

A version of this concerning accessible categories is provided in [7][Lemma
5.1]. The corresponding statement for locally finitely presentable categories
is an automatic consequence of the existence of colimits in functors
categories $\mathcal{B}^{D}$ where they are pointwise.

In particular, for a locally finitely presentable category, the arrow category
$\mathcal{B}^{2}$ is locally finitely presentable and we have

$(\mathcal{B}^{2})_{\omega}\simeq(\mathcal{B}_{\omega})^{2}$

Now we can turn to the explicit description of the following generator of
finitely presented objects in the coslices. We first introduce our candidate -
whose choice is of course guessed from a similar technique in [2] and [5].
Then we prove it to enjoy some lifting properties, stating that any finite
diagram (and in particular any triangle) in it can be obtained as a pushout of
a diagram of finitely presented objects of the same shape. From this we deduce
it to be closed under finite limits - in particular under retracts. Then we
prove the coslice to be locally finitely presentable and our candidate to be
its generator of finitely presented objects.

######  Definition 2.4.

For any object $B$ in $\mathcal{B}$, define the _coslice generator at $B$_ as
full subcategory $\mathcal{G}_{B}$ of $B\downarrow\mathcal{B}$ consisting of
morphisms $l:B\rightarrow C$ such that there exists some $k:K\rightarrow
K^{\prime}$ in $\mathcal{B}^{2}_{\omega}$ and $a:K\rightarrow B$ exhibiting
$n$ as the pushout

${{K}}$${{K^{\prime}}}$${{B}}$${{C}}$$\scriptstyle{a}$$\scriptstyle{k}$$\scriptstyle{l}$${\lrcorner}$

The first thing which is clear about this category is the following:

######  Lemma 2.5.

Objects of $\mathcal{G}_{B}$ are finitely presented in
$B\downarrow\mathcal{B}$

###### Proof.

Let be $F:I\rightarrow B\downarrow\mathcal{B}$ a filtered diagram; then form
what was said about filtered colimits in the coslice, we have ${\rm
cod}({\textup{colim}}\,F)\simeq{\textup{colim}}\,{\rm cod}\,F$. Then for any
situation as below

${K}$${{K^{\prime}}}$${B}$${{b_{*}K^{\prime}}}$${{{\textup{colim}}\,{\rm
cod}\,F}}$$\scriptstyle{b}$$\scriptstyle{b_{*}k}$$\scriptstyle{k}$$\scriptstyle{b^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{{\textup{colim}}\,F}$$\scriptstyle{a}$

the composite arrow $ak_{*}b:K^{\prime}\rightarrow{\textup{colim}}\,{\rm
cod}\,F$ lifts through some $q_{i}:{\rm
cod}\,F(i)\rightarrow{\textup{colim}}\,{\rm cod}\,F(i)$ as

${{K^{\prime}}}$${{b_{*}K^{\prime}}}$${{{\rm
cod}\,F(i^{\prime})}}$${{{\textup{colim}}\,{\rm
cod}\,F}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{q_{i}}$$\scriptstyle{\overline{a}}$$\scriptstyle{a}$

However, it is still not clear that the induced parallel lifts
$F(i)b:K\rightarrow F(i)$ and $\overline{a}k:K\rightarrow F(i)$ commute: but
from $K$ is finitely presented and $I$ is filtered, we know they are equalized
by some $F(d)$ for some morphism $d:i\rightarrow i^{\prime}$ in $I$, and
moreover, $F(i^{\prime})=F(d)F(i)$; then the universal property of the pushout
induces a universal map

${{K^{\prime}}}$${K}$${{b_{*}K^{\prime}}}$${{{\rm
cod}\,F(i^{\prime})}}$${B}$${{{\textup{colim}}\,{\rm
cod}\,F}}$$\scriptstyle{b}$$\scriptstyle{b_{*}k}$$\scriptstyle{k}$$\scriptstyle{b^{\prime}}$$\scriptstyle{{\textup{colim}}\,F}$$\scriptstyle{q_{i}}$$\scriptstyle{F(d)\overline{a}}$$\scriptstyle{a}$$\scriptstyle{F(i^{\prime})}$$\scriptstyle{\langle
F(i^{\prime}),F(d)\overline{a}\rangle}$$\scriptstyle{\lrcorner}$

Similarly, we use finitely presentedness of $K^{\prime}$ to prove that any two
lifts of $a$ have to be equalized by a further refinement. This proves the
pushout map $b_{*}k$ to be finitely presented in $B\downarrow\mathcal{B}$ ∎

Before being able to justify the appellation of generator, we must first
establish a series of technical properties allowing us to lift arrow and
finite diagrams from $B\downarrow\mathcal{B}$ to $\mathcal{B}_{\omega}$. Those
results will depend on the following useful technical lemma, which seems to
have first appeared in [2][sub-lemma 12]:

######  Lemma 2.6.

Let be a diagram as below

${{K_{0}}}$${{K_{0}^{\prime}}}$${K}$${B}$${C}$$\scriptstyle{k}$$\scriptstyle{n}$$\scriptstyle{a}$$\scriptstyle{a_{0}}$$\scriptstyle{\lrcorner}$

with $K,\,K_{0},\,K_{0}^{\prime}$ finitely presented and $k$ in
$\mathcal{B}^{2}_{\omega}$: then there exists a factorization

${{K_{0}}}$${{K_{1}}}$${B}$$\scriptstyle{a_{2}}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{0}}$

such that $a$ factorizes through the following pushout

${{K_{0}}}$${{K_{0}^{\prime}}}$${{K_{1}}}$${{{a_{2}}_{*}K_{0}^{\prime}}}$${K}$${B}$${C}$$\scriptstyle{k}$$\scriptstyle{b_{0}}$$\scriptstyle{a_{1}}$$\scriptstyle{{a_{2}}_{*}k}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{a}$

######  Remark.

Observe that from the properties of finitely presented objects, we can say
moreover that for any two parallel lifts of the same $a$

${{K_{1}}}$${{{a_{2}}_{*}K_{0}^{\prime}}}$${K}$${B}$${D}$$\scriptstyle{a_{1}}$$\scriptstyle{{a_{2}}_{*}k}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$$\scriptstyle{a}$$\scriptstyle{m^{\prime}}$$\scriptstyle{m}$

there exists a further factorization $a_{1}=a_{2}b_{1}$ with
$a_{2}:K_{2}\rightarrow B$ such that $m$ and $m^{\prime}$ are equalized by the
intermediate arrow

$(a_{2*}k)b_{1}m=(a_{2*}k)b_{1}m^{\prime}$

This lemma is also crucial to the following fullness-like property of the
$\mathcal{G}_{B}$ in the coslice, which allows to exhibit any arrow between
objects of the etale generator as a pushout square of finitely presented etale
map:

######  Lemma 2.7.

For any triangle in $B\downarrow\mathcal{B}$

${B}$${{C_{1}}}$${{C_{2}}}$$\scriptstyle{n_{1}}$$\scriptstyle{n_{2}}$$\scriptstyle{n}$

with $n_{1}$, $n_{2}$ in $\mathcal{G}_{B}$, there exists a triangle

${K}$${{K_{1}}}$${{K_{2}}}$$\scriptstyle{m_{1}}$$\scriptstyle{m_{2}}$$\scriptstyle{m}$

in $\mathcal{B}^{2}_{\omega}$ and some $a:K\rightarrow B$ such that all
squares below are pushouts

${K}$${{K_{1}}}$${B}$${{C_{1}}}$${{K_{2}}}$${{C_{2}}}$$\scriptstyle{m_{1}}$$\scriptstyle{m}$$\scriptstyle{a}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\scriptstyle{n_{1}}$$\scriptstyle{n}$$\scriptstyle{n_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{m_{2}}$

(so that in particular $n$ is in $\mathcal{G}_{C_{1}}$).

###### Proof.

As $n_{1}$ and $n_{2}$ are supposed in $\mathcal{G}_{B}$, they are induced
from pushouts as below

${{K_{1}}}$${{K_{1}^{\prime}}}$${K}$${B}$${{C_{1}}}$${{K_{2}^{\prime}}}$${{C_{2}}}$$\scriptstyle{m_{1}}$$\scriptstyle{a_{1}}$$\scriptstyle{n_{1}}$$\scriptstyle{n}$$\scriptstyle{n_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{a_{2}}$$\scriptstyle{m_{2}}$$\scriptstyle{\lrcorner}$

Now by filteredness of $\mathcal{B}_{\omega}\downarrow B$ there exists
$a_{3}:K_{3}\rightarrow B$ and a factorization

${{K_{1}}}$${{K_{3}}}$${B}$${{K_{2}}}$$\scriptstyle{b_{1}}$$\scriptstyle{a_{3}}$$\scriptstyle{b_{2}}$$\scriptstyle{a_{2}}$$\scriptstyle{a_{1}}$

and the pushouts themselves factorize

${{K_{1}}}$${{K_{1}^{\prime}}}$${{K_{2}}}$${{K_{3}}}$${{b_{1*}K_{1}^{\prime}}}$${{K^{\prime}_{2}}}$${{b_{2*}K_{2}^{\prime}}}$${B}$${{C_{1}}}$${{C_{2}}}$$\scriptstyle{a_{3}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{b_{1}}$$\scriptstyle{m_{1}}$$\scriptstyle{b_{1*}m_{1}}$$\scriptstyle{n_{2}}$$\scriptstyle{n_{1}}$$\scriptstyle{(b_{1*}m_{1})_{*}a_{3}}$$\scriptstyle{b_{2}}$$\scriptstyle{m_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

But now we can apply lemma 2.6 to the following situation (where $b_{1*}$
still is finitely presented as $K_{3}$ is)

${{b_{1*}K_{1}^{\prime}}}$${{K_{3}}}$${{b_{2*}K_{2}^{\prime}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{a_{3}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{n_{2}}$$\scriptstyle{(b_{1*}m_{1})_{*}a_{3}}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$

to exhibit a further factorization

${{K_{3}}}$${{b_{2*}K_{2}^{\prime}}}$${{b_{1*}K_{1}^{\prime}}}$${{K_{4}}}$${{b_{3*}b_{2*}K_{2}^{\prime}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{n_{2}}$$\scriptstyle{(b_{1*}m_{1})_{*}a_{3}}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$$\scriptstyle{a_{4}}$$\scriptstyle{b_{3}}$$\scriptstyle{b_{3*}b_{2*}m_{2}}$$\scriptstyle{m}$$\scriptstyle{\lrcorner}$

However, we cannot infer at this step that $m$ commutes with the other part of
the diagram. Indeed, one cannot infer that $mb_{1*}m_{1}$ and
$(b_{2*}m_{2})_{*}b_{3*}b_{2*}m_{2}$ commute. However they are equalized by
$n_{2*}a_{4}$, which provides two parallel lifts of the same situation

${{K_{3}}}$${{b_{1*}K_{1}^{\prime}}}$${{K_{3}}}$${{b_{2*}K_{2}^{\prime}}}$${{K_{4}}}$${{b_{3*}b_{2*}K_{2}^{\prime}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{n_{2}}$$\scriptstyle{(b_{1*}m_{1})_{*}a_{3}}$$\scriptstyle{n}$$\scriptstyle{\lrcorner}$$\scriptstyle{a_{4}}$$\scriptstyle{m}$$\scriptstyle{b_{1*}m_{1}}$$\scriptstyle{a_{3}}$$\scriptstyle{(b_{2*}m_{2})_{*}a_{3}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{b_{3*}b_{2*}m_{2}}$

so there exists a further factorization $a_{4}=a_{5}b_{4}$ with
$a_{5}:K_{5}\rightarrow B$ such that

$\displaystyle(b_{4*}b_{3*}b_{2*}m_{2})_{*}b_{4}mb_{1*}m_{1}$
$\displaystyle=(b_{4*}b_{3*}b_{2*}m_{2})_{*}b_{4}(b_{2*}m_{2})_{*}b_{3*}b_{2*}m_{2}$
$\displaystyle=b_{4*}b_{3*}b_{2*}m_{2}b_{4}b_{3}$

And now by the universal property of the pushout we have an arrow $m^{\prime}$
as in the diagram below

${{K_{3}}}$${{b_{1*}K_{1}^{\prime}}}$${{b_{2*}K_{2}^{\prime}}}$${{K_{5}}}$${{b_{4*}b_{3*}b_{1*}K_{1}^{\prime}}}$${{b_{3*}b_{2*}K_{2}^{\prime}}}$${B}$${{C_{1}}}$${{C_{2}}}$$\scriptstyle{n_{2}}$$\scriptstyle{n}$$\scriptstyle{a_{5}}$$\scriptstyle{b_{4*}b_{3*}b_{2*}m_{2}}$$\scriptstyle{n_{1}}$$\scriptstyle{b_{4*}b_{3*}b_{1*}m_{1}}$$\scriptstyle{\lrcorner}$$\scriptstyle{b_{4}b_{3}}$$\scriptstyle{b_{2*}m_{2}}$$\scriptstyle{b_{1*}m_{1}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{(b_{4*}b_{3*}b_{2*}m_{2})_{*}b_{4}m}$$\scriptstyle{m^{\prime}}$

Combining stability under pushouts and right cancellation of maps in
$\mathcal{B}^{2}_{\omega}$, we know $m^{\prime}$ to be in
$\mathcal{B}^{2}_{\omega}$; moreover, by right cancellation of pushout
squares, the right, bottom square is also a pushout, so that $n$ is exhibited
as a pushout of $m^{\prime}$ along the canonical inclusion
$(b_{3*}(b_{1*}m_{1}))_{*}a_{3}$. ∎

In particular, the following says we can lift any finite diagram in the etale
generator of $B$ into a diagram of the same shape made of finitely presented
etale arrows, from which it can be induced by pushout:

######  Lemma 2.8.

For any finite diagram $F:I\rightarrow\mathcal{G}_{B}$, there is some
$a:K\rightarrow B$ and some lifts

${{\mathcal{G}_{K}}}$${I}$${{\mathcal{G}_{B}}}$$\scriptstyle{a_{*}}$$\scriptstyle{\overline{F}}$$\scriptstyle{F}$$\scriptstyle{\simeq}$

where $a_{*}$ is the pushout functor. In particular the transition morphisms
of $F$ are obtained as pushouts of the corresponding transition morphisms of
$\overline{F}$.

###### Proof.

We saw that one can lift morphisms. Here we prove that one can lift finite
discrete diagrams and parallel pairs. Let be a discrete set
$(n_{i}:B\rightarrow C_{i})_{i\in I}$ with $I$ finite, with
$n_{i}=a_{i*}k_{i}$ and $k_{i}:K_{i}\rightarrow K_{i}^{\prime}$. Then, for
$\mathcal{B}_{\omega}\downarrow B$ is filtered, there exists some
$a:K\rightarrow B$ and for each $i\in I$ an arrow $b_{i}:K_{i}\rightarrow K$
such that $a_{i}=ab_{i}$; then in $\mathcal{G}_{K}$ one gets the following
diagram over $(n_{i})_{i\in I}$

${{b_{i*}K_{i}^{\prime}}}$${K}$${{b_{j*}K_{j}^{\prime}}}$$\scriptstyle{b_{i*}k_{i}}$$\scriptstyle{b_{j*}k_{j}}$$\scriptstyle{\ddots}$

Now consider a parallel pair

${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{n_{1}}$$\scriptstyle{n_{2}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

Then from lemma 2.7 there are respectively two lifts

${{K_{1}}}$${K}$${{K_{2}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{n_{1}}$$\scriptstyle{n_{2}}$$\scriptstyle{f}$$\scriptstyle{a}$$\scriptstyle{k_{1}}$$\scriptstyle{l}$$\scriptstyle{\lrcorner}$$\scriptstyle{k_{2}}$$\scriptstyle{\lrcorner}$
${{K_{1}^{\prime}}}$${{K^{\prime}}}$${{K_{2}^{\prime}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{n_{1}}$$\scriptstyle{n_{2}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{k_{1}^{\prime}}$$\scriptstyle{l^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{k_{2}^{\prime}}$$\scriptstyle{\lrcorner}$

Now one can find a common refinement
$a^{\prime\prime}:K^{\prime\prime}\rightarrow B$, $b:K\rightarrow
K^{\prime\prime}$ and $b^{\prime}:K^{\prime}\rightarrow K^{\prime\prime}$ of
$a$, $a^{\prime}$. Moreover, by applying upstream lemma 2.6 we can chose this
common refinement to be such that there exists a factorization

${{K^{\prime}}}$${{K_{1}^{\prime}}}$${{K^{\prime\prime}}}$${{b_{*}K^{\prime}}}$${B}$${{C_{1}}}$$\scriptstyle{n_{1}}$$\scriptstyle{k_{1}^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{a^{\prime\prime}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{c}$$\scriptstyle{b_{*}k_{1}}$

Now we can push the arrow $l^{\prime}$ along $c$ to get a diagram as below

${{K^{\prime}_{1}}}$${{b_{*}K_{1}}}$${{K^{\prime}_{2}}}$${{K^{\prime\prime}}}$${{b_{*}K_{2}}}$${{c_{*}K_{2}^{\prime}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{b_{*}k_{1}}$$\scriptstyle{c_{*}l^{\prime}}$$\scriptstyle{b_{*}l}$$\scriptstyle{a^{\prime\prime}}$$\scriptstyle{n_{2}}$$\scriptstyle{(b_{*}k_{2})_{*}a^{\prime\prime}}$$\scriptstyle{c}$$\scriptstyle{l^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{(b_{*}k_{1})_{*}a^{\prime\prime}}$$\scriptstyle{n_{1}}$$\scriptstyle{f}$$\scriptstyle{\scriptsize{(c_{*}l^{\prime})_{*}((b_{*}k_{1})_{*}a^{\prime\prime})}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{b_{*}k_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

where, by cancellation of pushouts, we have that

$f^{\prime}=((b_{*}k_{1})_{*}a^{\prime\prime})_{*}(c_{*}l^{\prime})$

Finally, again by lemma 2.6, we can get a last factorization
$a^{\prime\prime\prime}:K^{\prime\prime\prime}\rightarrow B$ and
$b^{\prime\prime}:K^{\prime\prime}\rightarrow K^{\prime\prime\prime}$ and a
factorization as below

${{b_{*}K_{1}}}$${{K^{\prime\prime}}}$${{b_{*}K_{2}}}$${{c_{*}K_{2}^{\prime}}}$${{b^{\prime\prime}_{*}b_{*}K_{1}}}$${{K^{\prime\prime\prime}}}$${{b^{\prime\prime}_{*}b_{*}K_{2}}}$${{C_{1}}}$${B}$${{C_{2}}}$$\scriptstyle{b_{*}k_{1}}$$\scriptstyle{c_{*}l^{\prime}}$$\scriptstyle{b_{*}l}$$\scriptstyle{n_{2}}$$\scriptstyle{n_{1}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{b^{\prime\prime}}$$\scriptstyle{a^{\prime\prime\prime}}$$\scriptstyle{(b_{*}k_{1})_{*}b^{\prime\prime}}$$\scriptstyle{b^{\prime\prime}_{*}b_{*}k_{1}}$$\scriptstyle{d}$$\scriptstyle{(b_{*}k_{2})_{*}a^{\prime\prime}}$$\scriptstyle{\scriptsize((b_{*}k_{1})_{*}b^{\prime\prime})_{*}b_{*}l}$$\scriptstyle{b^{\prime\prime}_{*}b_{*}k_{2}}$$\scriptstyle{b_{*}k_{2}}$$\scriptstyle{(c_{*}l^{\prime})_{*}((b_{*}k_{1})_{*}a^{\prime\prime})}$

But now, the pair $(dc_{*}l^{\prime},b^{\prime\prime}_{*}b_{*}k_{2})$ induces
a unique arrow

${{b^{\prime\prime}_{*}b_{*}K_{1}}}$${{b^{\prime\prime}_{*}bK_{2}}}$$\scriptstyle{\langle
dc_{*}l^{\prime},b^{\prime\prime}_{*}b_{*}k_{2}\rangle}$

which moreover satisfies $\langle
dc_{*}l^{\prime},b^{\prime\prime}_{*}b_{*}k_{2}\rangle
b^{\prime\prime}_{*}b_{*}k_{1}=b^{\prime\prime}_{*}b_{*}k_{2}$, so that by
cancellation of pushouts together with the pushout expression of $n_{1}$,
$n_{2}$, we have

$f^{\prime}=((b^{\prime\prime}_{*}b_{*}k_{1})_{*}a^{\prime\prime\prime})_{*}\langle
dc_{*}l^{\prime},b^{\prime\prime}_{*}b_{*}k_{2}\rangle$

Hence the parallel pair in $\mathcal{G}_{k^{\prime\prime\prime}}$

${{b^{\prime\prime}_{*}b_{*}k_{1}}}$${{b^{\prime\prime}_{*}b_{*}k_{2}}}$$\scriptstyle{\scriptsize((b_{*}k_{1})_{*}b^{\prime\prime})_{*}b_{*}l}$$\scriptstyle{\langle
dc_{*}l^{\prime},b^{\prime\prime}_{*}b_{*}k_{2}\rangle}$

is a lift of the parallel pair $f,f^{\prime}$ as desired. ∎

######  Corollary 2.9.

The coslice generator $\mathcal{G}_{B}$ is closed under finite colimits in the
cocomma $B\downarrow\mathcal{B}$.

###### Proof.

This is a consequence of the previous results, as we saw we can lift both
finite discrete diagrams and parallel pairs to some $\mathcal{G}_{K}$, where
we can compute the corresponding finite colimit as in
$\mathcal{B}^{2}_{\omega}$. Formally, for any finite diagram
$F:I\rightarrow\mathcal{G}_{B}$, choose a lift
$\overline{F}:I\rightarrow\mathcal{G}_{K}$ as provided by lemma 2.8. As $K$ is
finitely presented, $\mathcal{G}_{K}$ is a full subcategory of
$\mathcal{B}^{2}_{\omega}$ closed under finite colimits in
$\overrightarrow{\mathcal{B}_{\omega}}$, so that we have in each $i$ in $I$ a
diagram in $\mathcal{B}^{2}_{\omega}$

${{{\rm cod}(\overline{F}(i))}}$${K}$${{{\rm
cod}({\textup{colim}}\,\overline{F})}}$$\scriptstyle{{\textup{colim}}\,\overline{F}}$$\scriptstyle{\overline{F}(i)}$$\scriptstyle{\overline{q}_{i}}$

so that the colimit inclusions $\overline{q}_{i}$ of $\overline{F}$ are in
$\mathcal{B}^{2}_{\omega}$. Then by commutation of pushouts with colimits, we
have

$a_{*}{\textup{colim}}\,\overline{F}\simeq{\textup{colim}}\,a_{*}\overline{F}\simeq{\textup{colim}}\,F$

Hence for each $i$ in $I$ we have a diagram as below

${{{\rm cod}(\overline{F}(i))}}$${K}$${{{\rm
cod}({\textup{colim}}\,\overline{F})}}$${{F(i)}}$${B}$${{{\rm
cod}({\textup{colim}}\,F)}}$$\scriptstyle{\overline{F}(i)}$$\scriptstyle{\overline{q}_{i}}$$\scriptstyle{a}$$\scriptstyle{\overline{F}(i)_{*}a}$$\scriptstyle{F(i)}$$\scriptstyle{\lrcorner}$$\scriptstyle{q_{i}}$$\scriptstyle{{\textup{colim}}\,F}$$\scriptstyle{\lrcorner}$$\scriptstyle{{\textup{colim}}\,\overline{F}}$

where the front square is a pushout, exhibiting ${\textup{colim}}\,F$ as an
object of $\mathcal{G}_{B}$. Moreover, the colimit inclusions $q_{i}$ are
obtained as the pushouts

$q_{i}=(\overline{F}(i)_{*}a)_{*}(\overline{q}_{i})$

All of this suffice to proves that $\mathcal{G}_{B}$ is finitely cocomplete
and closed under finite colimits in $B\downarrow\mathcal{B}$. ∎

######  Lemma 2.10.

$\mathcal{G}_{B}$ is closed under the formation of retracts

###### Proof.

The crucial argument of this proof will be a lift endomorphisms to
endomorphisms. Let be $l:B\rightarrow C$ be a retract of a pushout $a_{*}k$ as
below

${{K^{\prime}}}$${K}$${C}$${{a_{*}K^{\prime}}}$${B}$${C}$$\scriptstyle{l}$$\scriptstyle{a}$$\scriptstyle{k}$$\scriptstyle{\lrcorner}$$\scriptstyle{s}$$\scriptstyle{r}$$\scriptstyle{l^{\prime}}$$\scriptstyle{a_{*}k}$

Then consider the corresponding idempotent $e$ in $B\downarrow\mathcal{B}$

${{a_{*}K^{\prime}}}$${{a_{*}K^{\prime}}}$${B}$${C}$$\scriptstyle{r}$$\scriptstyle{l^{\prime}}$$\scriptstyle{a_{*}k}$$\scriptstyle{a_{*}k}$$\scriptstyle{s}$$\scriptstyle{e}$

Then by lemma 2.6 we know there exists a factorization of pushout squares

${K}$${{K^{\prime}}}$${{K^{\prime}}}$${{K_{0}}}$${{b_{0*}K^{\prime}}}$${a_{*}K^{\prime}}$${B}$${{a_{*}K^{\prime}}}$$\scriptstyle{a_{*}k}$$\scriptstyle{a_{0}}$$\scriptstyle{b_{0*}k}$$\scriptstyle{b_{0}}$$\scriptstyle{k}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{e}$$\scriptstyle{n_{*}a}$$\scriptstyle{l}$

Then the universal property of the pushout induces a unique map $l$ as below

${{K^{\prime}}}$${{b_{0*}K^{\prime}}}$${K}$${{a_{*}K^{\prime}}}$${{K_{0}}}$${{b_{0*}K^{\prime}}}$${B}$${{a_{*}K^{\prime}}}$$\scriptstyle{a_{*}k}$$\scriptstyle{a_{0}}$$\scriptstyle{n_{*}a}$$\scriptstyle{l}$$\scriptstyle{a_{*}k}$$\scriptstyle{b_{0}}$$\scriptstyle{k}$$\scriptstyle{l_{0}}$$\scriptstyle{\lrcorner}$$\scriptstyle{b_{0*}k}$$\scriptstyle{b_{0*}k}$

Though this endomorphism $l:b_{0*}K^{\prime}\rightarrow b_{0*}K^{\prime}$ is
not necessarily an idempotent, we can still compute the coequalizer

${{b_{0*}K^{\prime}}}$${{K_{0}}}$${{b_{0*}K^{\prime}}}$${{{\textup{coeq}}(l,1_{b_{0*}K^{\prime}})}}$$\scriptstyle{l}$$\scriptstyle{b_{0*}k}$$\scriptstyle{b_{0*}k}$$\scriptstyle{1_{b_{0*}K^{\prime}}}$$\scriptstyle{q_{l,1_{b_{0*}K^{\prime}}}b_{0*}k}$$\scriptstyle{q_{l,1_{b_{0*}K^{\prime}}}}$

which where $q_{l,1_{b_{0*}K^{\prime}}}$ still is in
$\mathcal{B}^{2}_{\omega}$ by closure of $\mathcal{B}_{\omega}$ under finite
colimits. Then by commutation of coequalizers with pushouts, the right square
in the following diagram is a pushout

${{b_{0*}K^{\prime}}}$${{b_{0*}K^{\prime}}}$${{{\textup{coeq}}(l,1_{b_{0*}K^{\prime}})}}$${{a_{*}K^{\prime}}}$${{a_{*}K^{\prime}}}$${{{\textup{coeq}}(e,1_{a_{*}K^{\prime}})}}$$\scriptstyle{l}$$\scriptstyle{1_{b_{0*}K^{\prime}}}$$\scriptstyle{q_{l,1_{b_{0*}K^{\prime}}}}$$\scriptstyle{(b_{0*}k)_{*}a_{0}}$$\scriptstyle{1_{a_{*}K^{\prime}}}$$\scriptstyle{e}$$\scriptstyle{q_{e,1_{a_{*}K^{\prime}}}}$$\scriptstyle{\lrcorner}$

But as $e$ was induced from the pair $(r,s)$, $(r,s)$ is in turn a splitting
of the idempotent $e$, so that $r$ happens to coincide with the coequalizer
above, that is

$C\simeq{\textup{coeq}}(e,1_{a_{*}K^{\prime}})\hskip
28.45274pt\textrm{and}\hskip 28.45274ptr=q_{e,1_{a_{*}K^{\prime}}}$

Then by composition of pushouts, $n$ itself is exhibited as the following
pushout of a map in $\mathcal{B}^{2}_{\omega}$:

${{b_{0*}K^{\prime}}}$${{K_{0}}}$${{{\textup{coeq}}(l,1_{b_{0*}K^{\prime}})}}$${{a_{*}K^{\prime}}}$${B}$${C}$$\scriptstyle{q_{l,1_{b_{0*}K^{\prime}}}}$$\scriptstyle{(b_{0*}k)_{*}a_{0}}$$\scriptstyle{q_{e,1_{a_{*}K^{\prime}}}}$$\scriptstyle{\lrcorner}$$\scriptstyle{b_{0*}k}$$\scriptstyle{a_{0}}$$\scriptstyle{a_{0*}k}$$\scriptstyle{\lrcorner}$$\scriptstyle{n}$$\scriptstyle{q_{l,1_{b_{0*}K^{\prime}}}b_{0*}k}$

This proves that the category $\mathcal{G}_{B}$ is closed under retracts. ∎

######  Remark.

In fact we could have already guessed that $\mathcal{G}_{B}$ had to be closed
under retracts, for it was proven before to be closed under finite colimits.
However we think it is worth emphasizing why retracts of morphisms in
$\mathcal{G}_{B}$ are still in $\mathcal{G}_{B}$, as someone proving first
that $\mathcal{G}_{B}$ is a dense generator would let think at first sight
that one should also take retracts of objects of $\mathcal{G}_{B}$ to have all
the finitely presented objects - which would make everything far more
complicated when applying the results for concrete situations: it is somewhat
reassuring to see precisely why this hindrance is illusory.

######  Remark.

In [6][Warning 2.2.5], we are warned that in the context of
$(\infty,1)$-categories, the statement above ceases to be true: the analog of
$\mathcal{G}_{B}$ are not anymore closed under retract. But we think this is
due to the fact that, unlike in 1-categories, splitting of idempotents in
$(\infty,1)$-categories ceases to be constructible by mean of finite
(co)limits, so that in this context there is no analog to our argument
involving expression of the splitting as a coequalizer.

######  Lemma 2.11.

$\mathcal{G}_{B}$ is a dense generator of $B\downarrow\mathcal{B}$.

###### Proof.

From what was said before on the arrow category, we know that any
$f:B\rightarrow C$ decomposes as a filtered colimit
$f\simeq{\textup{colim}}\,\mathcal{B}^{2}_{\omega}\downarrow f$ in the arrow
category $\mathcal{B}^{2}$. Moreover, from computation of colimits in
$\mathcal{B}^{2}$, we have both that

$B\simeq\underset{(k,b,b^{\prime})\in\mathcal{B}^{2}_{\omega}\downarrow
f}{{\textup{colim}}}{\rm dom}(k)\hskip
28.45274ptC\simeq\underset{(k,b,b^{\prime})\in\mathcal{B}^{2}_{\omega}\downarrow
f}{{\textup{colim}}}{\rm cod}(k)$

where the colimits range over the squares

${K}$${{K^{\prime}}}$${B}$${C}$$\scriptstyle{b}$$\scriptstyle{k}$$\scriptstyle{b^{\prime}}$$\scriptstyle{f}$

But each of those squares induces uniquely an arrow from the pushout

${K}$${{K^{\prime}}}$${B}$${{b_{*}K^{\prime}}}$${C}$$\scriptstyle{b}$$\scriptstyle{k}$$\scriptstyle{b^{\prime}}$$\scriptstyle{f}$$\scriptstyle{b_{*}k}$$\scriptstyle{\lrcorner}$$\scriptstyle{\langle
f,b^{\prime}\rangle}$

while a morphism in $\mathcal{B}^{2}_{\omega}$

${{K_{1}^{\prime}}}$${{K_{2}^{\prime}}}$${{K_{1}}}$${{K_{2}}}$${C}$${B}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$$\scriptstyle{k_{1}}$$\scriptstyle{f}$$\scriptstyle{b_{1}^{\prime}}$$\scriptstyle{b_{2}^{\prime}}$$\scriptstyle{k_{2}}$$\scriptstyle{l}$$\scriptstyle{l^{\prime}}$

induces a morphism between the pushouts in the coslice

${{b_{1*}K^{\prime}_{1}}}$${{b_{2*}K^{\prime}_{2}}}$${B}$${C}$$\scriptstyle{f}$$\scriptstyle{\langle
f,b_{1}^{\prime}\rangle}$$\scriptstyle{\langle
f,b_{2}^{\prime}\rangle}$$\scriptstyle{\langle
f,k_{2*}b_{2}l^{\prime}\rangle}$$\scriptstyle{b_{1*}k_{1}}$$\scriptstyle{b_{2*}k_{2}}$

This defines a “pushout” functor

${{\mathcal{B}^{2}_{\omega}\downarrow f}}$${{\mathcal{G}_{B}\downarrow
f}}$$\scriptstyle{\langle f,-\rangle}$

sending $(k,b,b^{\prime})$ on the induced $\langle
f,b^{\prime}\rangle:b_{*}k\rightarrow f$ in the coslice
$B\downarrow\mathcal{B}$. From lemma 2.7, we know that the pushout functor is
full; moreover it is essentially surjective by the very definition of
$\mathcal{G}_{B}$, and as $\mathcal{B}^{2}_{\omega}\downarrow f$ is filtered,
lemma 1.6 ensures us that $\langle f,-\rangle$ is cofinal and that
$\mathcal{G}_{B}\downarrow f$ is moreover filtered - though the later item can
also easily be checked concretely.

But now we have a pseudocommutative square of functors

${{\mathcal{B}^{2}_{\omega}\downarrow f}}$${{\mathcal{G}_{B}\downarrow
f}}$${{\mathcal{B}^{2}}}$${{B\downarrow\mathcal{B}}}$$\scriptstyle{\langle
f,-\rangle}$$\scriptstyle{{\rm
dom}}$$\scriptstyle{\pi}$$\scriptstyle{\iota_{B}}$$\scriptstyle{\simeq}$

where $\pi:\mathcal{G}_{B}\downarrow f\rightarrow B\downarrow\mathcal{B}$ is
the projection sending the triangle

${K}$${{K^{\prime}}}$${B}$${{b_{*}K^{\prime}}}$${C}$$\scriptstyle{b}$$\scriptstyle{b_{*}k}$$\scriptstyle{k}$$\scriptstyle{b^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{f}$$\scriptstyle{a}$

on $b_{*}k:B\rightarrow b_{*}K^{\prime}$ in $\mathcal{B}$. From $\langle
f,-\rangle$ is cofinal, we have an isomorphism of the corresponding filtered
colimit in $\mathcal{B}^{2}$

$\displaystyle f$
$\displaystyle\simeq\underset{\mathcal{B}^{2}_{\omega}\downarrow
f}{{\textup{colim}}}\;{\rm cod}$
$\displaystyle\simeq\underset{\mathcal{B}^{2}_{\omega}\downarrow
f}{{\textup{colim}}}\;\iota_{B}\pi\langle f,-\rangle$
$\displaystyle\simeq\underset{\mathcal{G}_{B}\downarrow
f}{{\textup{colim}}}\;\iota_{B}\pi$

But we saw that $\iota_{B}:B\downarrow\mathcal{B}\hookrightarrow\mathcal{B}$
creates filtered colimits: hence we already have a filtered colimit in
$B\downarrow\mathcal{B}$

$f\simeq{\textup{colim}}\;\mathcal{G}_{B}\downarrow f$

This proves that $\mathcal{G}_{B}$ is a dense generator in
$B\downarrow\mathcal{B}$ consisting of finitely presented objects. ∎

Putting altogether the previous lemma, we have proven the main result of this
part:

######  Theorem 2.12.

Let be $\mathcal{B}$ a locally finitely presentable category and $B$ an object
of $\mathcal{B}$. Then the coslice $B\downarrow\mathcal{B}$ is locally
finitely presentable and we have an equivalence

$(B\downarrow\mathcal{B})_{\omega}\simeq\mathcal{G}_{B}$

It is also worth precising that the codomain functor is actually part of this
locally finitely presentable structure:

######  Proposition 2.13.

The codomain functor ${\rm cod}:B\downarrow\mathcal{B}\rightarrow\mathcal{B}$
is the right part of a morphism of locally finitely presentable categories.

###### Proof.

From what was said in part 1, we know that ${\rm cod}$ is both finitary and
continuous. Its left adjoint ${\rm cod}^{*}$ can be defined as sending $C$ in
$\mathcal{B}$ to the coproduct inclusion $i_{B,C}^{B}:B\rightarrow B+C$ and a
morphism $f:C_{1}\rightarrow C_{2}$ on the morphism induced by universal
property of the coproduct. It is immediate that this functor preserves
finitely presented objects as for a finitely presented $K$ the following
square

${0}$${K}$${B}$${{B+K}}$$\scriptstyle{{\rm
cod}^{*}(K)}$$\scriptstyle{!_{B}}$$\scriptstyle{!_{K}}$$\scriptstyle{\lrcorner}$

is a pushout and $0$ is always finitely presented. ∎

Now observe that in the particular case of the coslice at finitely presented
objects, the descriptions above simplifies and we have

######  Corollary 2.14.

Let be $K$ a finitely presented object. Then we have

$(K\downarrow\mathcal{B})_{\omega}\simeq K\downarrow\mathcal{B}_{\omega}$

###### Proof.

This is because in this case $\mathcal{G}_{K}$ is a subcategory of
$\mathcal{B}^{2}_{\omega}$ as $\mathcal{B}_{\omega}$ is closed under finite
colimits, so that in any pushout square

${{K_{0}}}$${{K_{0}^{\prime}}}$${K}$${{k_{*}K_{0}^{\prime}}}$$\scriptstyle{k}$$\scriptstyle{k_{0}}$$\scriptstyle{k_{*}k_{0}}$$\scriptstyle{\lrcorner}$

the pushout $k_{*}K_{0}^{\prime}$ is finitely presented. Conversely any arrow
$k:K\rightarrow K^{\prime}$ in $\mathcal{B}_{\omega}^{2}$ is its own pushout
along the identity map of $K$. ∎

To finish, let us examine functoriality of the construction.

######  Proposition 2.15.

For $f:B_{1}\rightarrow B_{2}$ in $\mathcal{B}$, the adjoint pair

${{B_{2}\downarrow\mathcal{B}}}$${{B_{1}\downarrow\mathcal{B}}}$$\scriptstyle{f^{!}}$$\scriptstyle{f_{*}}$$\scriptstyle{\dashv}$

defines a morphisms $f\downarrow\mathcal{B}$ of locally finitely presentable
categories.

###### Proof.

Let us see why the right adjoint $f^{!}$, which is precomposition with $f$, is
continuous and finitary. Consider any limit in $B_{2}\downarrow\mathcal{B}$ of
a diagram $F:I\rightarrow B_{2}\downarrow\mathcal{B}$, knowing that this limit
is the induced map $\lim F=(F(i))_{i\in I}$. Then by naturality of the
universal property of the limit at $f$

${{\mathcal{B}[B_{2},\lim\,{\rm
cod}\,F]}}$${{\mathcal{B}^{I}[\Delta_{B_{2}},F]}}$${{\mathcal{B}[B_{,}\lim\,{\rm
cod}\,F]}}$${{\mathcal{B}^{I}[\Delta_{B_{1}},F]}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{B}^{I}[\Delta_{f},F]}$$\scriptstyle{f^{!}}$$\scriptstyle{\simeq}$

where $\Delta$ returns the constant diagram at an object and
$\mathcal{B}^{I}[\Delta_{f},F]$ is precomposition of a cone over $F$ with tip
$B_{2}$ with $f$, we know that $(F(i))_{i\in I}f$ is the universal map induced
from the composite cone $(F(i)f)_{i\in I}$, so that

$\displaystyle f^{!}(\lim\,F)$ $\displaystyle=(F(i))_{i\in I}f$
$\displaystyle=(F(i)f)_{i\in I}$ $\displaystyle=(f^{!}F(i))_{i\in I}$
$\displaystyle=\lim f^{!}F$

Concerning filtered colimits, recall that again for
$F:I\rightarrow\mathcal{B}$ filtered, ${\rm
cod}\,{\textup{colim}}\,F={\textup{colim}}\,{\rm cod}\,F$ and
${\textup{colim}}\,F$ is equal to any of the composite $q_{i}F(i)$ with
$q_{i}:{\rm cod}\,F(i)\rightarrow{\textup{colim}}\,{\rm cod}\,F(i)$ the
colimit inclusion. Hence it is immediate that for any $i$ in $I$ we have

$\displaystyle f^{!}({\textup{colim}}\,F)$
$\displaystyle={\textup{colim}}\,Ff$ $\displaystyle=q_{i}f_{i}f$
$\displaystyle={\textup{colim}}\,f^{!}F$

Finally, compositions of pushouts makes obvious that the left adjoint $f_{*}$
sends finitely presented objects of $B_{1}\downarrow\mathcal{B}$ to finitely
presented objects in $B_{2}\downarrow\mathcal{B}$. ∎

Observe that we have in particular a triangle in ${\bf LFP}$

${{B_{2}\downarrow\mathcal{B}}}$${{B_{1}\downarrow\mathcal{B}}}$${{\mathcal{B}}}$$\scriptstyle{f\downarrow\mathcal{B}}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{{\rm cod}}$$\scriptstyle{=}$

induced from the triangle

${{B_{2}}}$${{B_{1}}}$${0}$$\scriptstyle{f}$$\scriptstyle{!_{B_{2}}}$$\scriptstyle{!_{B_{1}}}$

This exhibits in particular the codomain functor ${\rm
cod}:B\downarrow\mathcal{B}\rightarrow\mathcal{B}$ as the transition morphism
$!_{B}\downarrow\mathcal{B}$.

The construction above defines a functor

${{\mathcal{B}^{{}^{\rm op}}}}$${{\bf
LFP}}$$\scriptstyle{(-)\downarrow\mathcal{B}}$

In a future work, we shall describe how this is related to a notion of
_spectral 2-site_ of a finite limit theory, in the context of a _2-dimensional
geometry_ associated to Gabriel-Ulmer duality.

##  3 Comma-objects in ${\bf LFP}$

In this section we analyse a similar problem, concerning this time the comma
of locally a finitely presentable category at a morphism of locally finitely
presentable categories. In [8][Proposition 6.1.1] and [1][Proposition 2.43] it
is proven that for any cospan of accessible functors between accessible
categories

${{\mathcal{A}_{2}}}$${{\mathcal{A}_{1}}}$${{\mathcal{B}}}$$\scriptstyle{F_{2}}$$\scriptstyle{F_{1}}$

the comma category $F_{1}\downarrow F_{2}$ is accessible; then in particular
whenever $F_{1}$ and $F_{2}$ are locally presentable, preservation of limits
makes the comma complete and then locally presentable itself. However, in both
of those sources, the strategy of the proof does not allow to control the rank
of accessibility of the comma, even when we know the ranks of the functors and
the categories involved, with for instance $\lambda$ a convenient cardinal for
all those ranks. The argument relies indeed on the fact that there exists
_another cardinal_ $\mu$ larger than $\lambda$ such that both $F_{1}$ and
$F_{2}$ moreover preserves $\mu$-finitely presented objects. Then one can
prove only $F_{1}\downarrow F_{2}$ to be $\mu$-accessible, with its
$\mu$-presentable objects being of the form $f:F_{1}(K_{1})\rightarrow
F_{2}(K_{2})$ with $K_{1}$ and $K_{2}$ being $\mu$-presentable in
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ respectively. However this is not
satisfactory for our purpose. While we do not investigate the general form of
the comma involving two functors, this section is devoted to improving this
result concerning comma of the form $F\downarrow\mathcal{B}$.

In this section, we prove that for a morphism of locally finitely presentable
categories $F:\mathcal{A}\rightarrow\mathcal{B}$ consisting of an adjoint pair
$F^{*}\dashv F_{*}$ with $F_{*}$ finitary, the comma object
$F_{*}\downarrow\mathcal{B}$ is locally _finitely_ presentable. The intuition
behind this claim is the following: from Gabriel-Ulmer duality, we know that
the 2-category ${\bf LFP}$ of locally finitely presentable categories is
equivalent to the opposite 2-category ${\bf Lex}^{{}^{\rm op}}$ of small lex
categories with lex functors between them. But ongoing investigation on
2-categorical model theory, as well as the work of [4], tell us that ${\bf
Lex}$ is a _locally finitely bipresentable 2-category_ \- or at least a
_locally presentable 2-category_ in [4] \- which indicate its closure under
both small pseudolimits and pseudocolimits. Hence if ${\bf Lex}$ is closed
under pseudocolimit, ${\bf LFP}$ must be closed under pseudolimits, in
particular under comma objects.

At first sight, such comma objects might be different from those computed in
${\bf Cat}$. If ${\bf Lex}$, being KZ-monadic on ${\bf Cat}$, is known to be
equipped with a 2-functor ${\bf Lex}\rightarrow{\bf Cat}$ preserving
pseudolimits, controlling its pseudocolimits is more difficult: by duality,
while controlling pseudocolimits of ${\bf LFP}$ should be easy, controlling
its pseudolimits is not.

However, we are going to see in this section that actually, not only the comma
computed in ${\bf Cat}$ is locally finitely presentable as well as all the
canonical functors involved, but moreover, that this comma object, as computed
in ${\bf Cat}$, also has the universal property of the comma object in ${\bf
LFP}$.

In ${\bf Cat}$, the is a weak algebraic factorization system

$(coreflection,\,fibration)$

which is obtained as follows. Let be $F:\mathcal{A}\rightarrow\mathcal{B}$ in
${\bf Cat}$. Then form the following comma objects

${{F\downarrow\mathcal{B}}}$${{\mathcal{B}}}$${{\mathcal{A}}}$${{\mathcal{B}}}$$\scriptstyle{F}$$\scriptstyle{1_{\mathcal{B}}}$$\scriptstyle{\pi_{F}}$$\scriptstyle{\pi_{\mathcal{B}}}$$\scriptstyle{\lambda_{F}}$

and observe that the pair $(1_{\mathcal{A}},f)$ induces in both cases a
canonical equality 2-cell

${{\mathcal{A}}}$${{\mathcal{B}}}$${{F\downarrow\mathcal{B}}}$$\scriptstyle{1_{F}}$$\scriptstyle{{\rm
cod}_{F}}$$\scriptstyle{F}$$\scriptstyle{=}$

where $1_{F}$ sends $A$ in $\mathcal{A}$ to the identity 1-cell
$F(1_{A})=1_{F(A)}$.

######  Remark.

Beware that the pair (coreflection, fibration) is not a strong orthogonality
structure, only a weak one. However the factorization above is canonical up to
equality, hence the qualificative of _algebraic_.

Now we apply this factorization to the direct image of a morphism of locally
finitely presentable categories. Let be $F:\mathcal{A}\rightarrow\mathcal{B}$
a ${\bf LFP}$ morphism, consisting of a finitary continuous functor
$F_{*}:\mathcal{A}\rightarrow\mathcal{B}$, with its corresponding left adjoint
$F^{*}$, which is known from lemma 2.2 to restrict to a functor
$F^{*}_{\omega}:\mathcal{B}_{\omega}\rightarrow\mathcal{A}_{\omega}$. Then one
can consider the (coreflection, fibration) factorization of $F_{*}$ in ${\bf
Cat}$

${{\mathcal{A}}}$${{\mathcal{B}}}$${{F_{*}\downarrow\mathcal{B}}}$$\scriptstyle{1_{F_{*}}}$$\scriptstyle{{\rm
cod}{F_{*}}}$$\scriptstyle{F_{*}}$$\scriptstyle{=}$

Since $F_{*}$ is itself finitary and continuous, we know that limits and
filtered colimits in $F_{*}\downarrow\mathcal{B}$ are computed in the arrow
category $\mathcal{B}$ as respectively

${{F(\underset{i\in I}{\lim}\,A_{i})}}$${{\underset{i\in
I}{\lim}\,B_{i}}}$${{F(A_{i})}}$${{B_{i}}}$$\scriptstyle{f_{i}}$$\scriptstyle{F(q_{i})}$$\scriptstyle{q^{\prime}_{i}}$$\scriptstyle{\underset{i\in
I}{\lim}\,f_{i}}$ ${{F(A_{i})}}$${{B_{i}}}$${{F(\underset{i\in
J}{{\textup{colim}}}\,A_{i})}}$${{\underset{i\in
J}{{\textup{colim}}}\,B_{i}}}$$\scriptstyle{f_{i}}$$\scriptstyle{F(q_{i})}$$\scriptstyle{q^{\prime}_{i}}$$\scriptstyle{\underset{i\in
J}{{\textup{colim}}}\,f_{i}}$

(whenever $J$ is filtered) with $F(\lim_{i\in I}A_{i})\simeq\lim_{i\in
I}F(A_{i})$ and $F({\textup{colim}}_{i\in J}A_{i})\simeq{\textup{colim}}_{i\in
J}F(A_{i})$.

As a consequence, ${\rm cod}_{F_{*}}$ is finitary and continuous, as well as
$1_{F_{*}(-)}$ and its pseudoretract $\pi_{F_{*}}$ sending
$f:F_{*}(A)\rightarrow B$ on $A$. From what follows we shall see they are
actually part of LFP morphisms.

Most of this section will be devoted to prove that
$F_{*}\downarrow\mathcal{B}$ is finitely presented. As in section 2, we first
guess a candidate for the generator for finitely presented objects, and prove
it to be effectively such. Then again proving it to be closed under retract
also proves it to contain all finitely presented objects. Finally, the
previous discussion above ensuring existence of limits in the comma, we will
know it to be locally finitely presented.

######  Definition 3.1.

Define the _generator of the comma_ as the full subcategory
$\mathcal{G}_{F_{*}}$ having as objects all the arrows $f:F_{*}(M)\rightarrow
B$ with $M$ a finitely presented object of $\mathcal{A}_{\omega}$ and such
that there exists a pushout square

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${B}$$\scriptstyle{a}$$\scriptstyle{k}$$\scriptstyle{a^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\lrcorner}$

######  Remark.

From both $\mathcal{A}_{\omega}$ and $\mathcal{B}_{\omega}$ are essentially
small, and $\mathcal{B}$ being locally small, $\mathcal{G}_{F_{*}}$ is also
essentially small.

######  Lemma 3.2.

$\mathcal{G}_{F_{*}}$ consists of finitely presented objects in
$F_{*}\downarrow\mathcal{B}$.

###### Proof.

Let be $f_{(-)}:I\rightarrow F_{*}\downarrow\mathcal{B}$ a filtered diagram,
with $f_{i}:F_{*}(A_{i})\rightarrow B_{i}$. Now let be a diagram as below:

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${s_{*}K^{\prime}}$${{F_{*}(\underset{i\in
I}{{\textup{colim}}}\,A_{i})}}$${{\underset{i\in
I}{{\textup{colim}}}\,B_{i}}}$$\scriptstyle{s}$$\scriptstyle{k}$$\scriptstyle{k_{*}s}$$\scriptstyle{s_{*}k}$$\scriptstyle{\lrcorner}$$\scriptstyle{F_{*}(a)}$$\scriptstyle{b}$$\scriptstyle{\underset{i\in
I}{{\textup{colim}}}\,f_{i}}$

Then for $M$ is finitely presented in $\mathcal{A}$ we have a lift for some
$i\in I$

${M}$${{A_{i}}}$${{\underset{i\in
I}{{\textup{colim}}}\,A_{i}}}$$\scriptstyle{\overline{a}}$$\scriptstyle{q_{i}}$$\scriptstyle{a}$

Moreover, this factorization also provides by precomposition with $a$ a lift
of $F_{*}(a)s$ witnessing finitely presentedness of $K$ relatively to the
filtered colimit $F_{*}({\textup{colim}}_{i\in
I}A_{i})\simeq{\textup{colim}}_{i\in I}F_{*}(A_{i})$:

${K}$${{F_{*}(M)}}$${{F_{*}(A_{i})}}$${{F_{*}(\underset{i\in
I}{{\textup{colim}}}\,A_{i})}}$$\scriptstyle{F_{*}(\overline{a})}$$\scriptstyle{F_{*}(q_{i})}$$\scriptstyle{F_{*}(a)}$$\scriptstyle{s}$

On the other side, the arrow $k:K\rightarrow K^{\prime}$ is finitely presented
in $\mathcal{B}^{2}$, and hence also lifts as follows for some $i^{\prime}\in
I$:

${K}$${{K^{\prime}}}$${{F_{*}(A_{i^{\prime}})}}$${{B_{i^{\prime}}}}$${{F_{*}(\underset{i\in
I}{{\textup{colim}}}\,A_{i})}}$${{\underset{i\in
I}{{\textup{colim}}}\,B_{i}}}$$\scriptstyle{t}$$\scriptstyle{k}$$\scriptstyle{t^{\prime}}$$\scriptstyle{f_{i^{\prime}}}$$\scriptstyle{F_{*}(q_{i^{\prime}})}$$\scriptstyle{q^{\prime}_{i^{\prime}}}$$\scriptstyle{\underset{i\in
I}{{\textup{colim}}}\,f_{i}}$$\scriptstyle{F_{*}(a)s}$$\scriptstyle{bk_{*}a}$

Hence we have two parallel lifts $(F_{*}(\overline{a})s,i)$ and
$(t,i^{\prime})$ of $F_{*}(a)s$, so by finite presentedness of $K$ we know
there exists some $j\in I$ and a common refinement $d:i\rightarrow j$,
$d^{\prime}:i^{\prime}\rightarrow j$ of those parallel lifts

${K}$${{F_{*}(A_{i})}}$${{F_{*}(A_{i^{\prime}})}}$${{F_{*}(A_{j})}}$${{F_{*}(\underset{i\in
I}{{\textup{colim}}}\,A_{i})}}$$\scriptstyle{F_{*}(f_{d})}$$\scriptstyle{t}$$\scriptstyle{F_{*}(f_{d^{\prime}})}$$\scriptstyle{F_{*}(\overline{a})s}$$\scriptstyle{F_{*}(q_{j})}$$\scriptstyle{F_{*}(q_{i})}$$\scriptstyle{F_{*}(q_{i^{\prime}})}$

Inserting this common refinement in the following diagram

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{F_{*}(A_{i^{\prime}})}}$${{B_{i}}}$${{F_{*}(A_{j})}}$${{B_{j}}}$$\scriptstyle{t}$$\scriptstyle{F_{*}(f_{d^{\prime}})}$$\scriptstyle{s}$$\scriptstyle{k}$$\scriptstyle{t^{\prime}}$$\scriptstyle{g_{d^{\prime}}}$$\scriptstyle{f_{j}}$$\scriptstyle{f_{i}}$$\scriptstyle{F_{*}(f_{d})F_{*}(\overline{a})}$

provides us with a commuting square equalizing $(a,k)$, inducing a
factorization through the pushout

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{s_{*}K^{\prime}}}$${{F_{*}(A_{j})}}$${{B_{j}}}$$\scriptstyle{s}$$\scriptstyle{k}$$\scriptstyle{f_{j}}$$\scriptstyle{F_{*}(f_{d})F_{*}(\overline{a})}$$\scriptstyle{s_{*}k}$$\scriptstyle{k_{*}s}$$\scriptstyle{g_{d^{\prime}}t^{\prime}}$$\scriptstyle{\overline{b}}$$\scriptstyle{\lrcorner}$

This provides a lift $((f_{d}\overline{a},\overline{b}),j)$ of $(a,b)$ in the
comma $F_{*}\downarrow\mathcal{B}$ as desired.

Now we must check that any two parallel lifts have a common refinement.
Suppose we have a situation as below

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{s_{*}K^{\prime}}}$${{F_{*}(A_{i})}}$${{\;B_{i}}}$${{F_{*}(A_{i^{\prime}})}}$${{B_{i^{\prime}}}}$${{F_{*}(\underset{i\in
I}{{\textup{colim}}}\,A_{i})}}$${{\underset{i\in
I}{{\textup{colim}}}\,B_{i}}}$$\scriptstyle{F_{*}(q_{i})}$$\scriptstyle{\underset{i\in
I}{{\textup{colim}}}\,f_{i}}$$\scriptstyle{F_{*}(a)}$$\scriptstyle{s_{*}k}$$\scriptstyle{b}$$\scriptstyle{f_{i}}$$\scriptstyle{q^{\prime}_{i}}$$\scriptstyle{f_{i^{\prime}}}$$\scriptstyle{q^{\prime}_{i^{\prime}}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{k}$$\scriptstyle{s}$$\scriptstyle{\lrcorner}$$\scriptstyle{F_{*}(a^{\prime})}$$\scriptstyle{F_{*}(q_{i^{\prime}})}$

where the two lifts $((a,b),i)$ and $((a^{\prime},b^{\prime}),i^{\prime})$
define the same arrow $s_{*}k\rightarrow{\textup{colim}}_{i\in I}f_{i}$ in
$F_{*}\downarrow\mathcal{B}$. Then from $M$ is finitely presentable, there is
a common refinement $d:i\rightarrow j$ and $d^{\prime}:i^{\prime}\rightarrow
j$ for the parallel lifts $a,a^{\prime}$ in $\mathcal{A}$. On the other hand
we also have a common refinement $e:i\rightarrow j^{\prime}$ and
$e^{\prime}:i^{\prime}\rightarrow j^{\prime}$ for the parallel lifts
$((F_{*}(a)s,bk*s),i)$ and $((F_{*}(a^{\prime})s,b^{\prime}k*s),i^{\prime})$
from $k$ to ${\textup{colim}}_{i\in I}f_{i}$ in $\mathcal{B}^{2}$. Moreover,
because the left part of those parallel lifts factorize through $s$, we are
provided with two further parallel lifts $(da=d^{\prime}a^{\prime},j)$ and
$(ea=ea^{\prime},j^{\prime})$ in $\mathcal{A}$, which have hence themselves a
further refinement $h:j\rightarrow l$, $h^{\prime}:j^{\prime}\rightarrow l$ in
$I$. Hence by a similar argument as above, this common refinement equalizes
$s$ and $k$, hence factorizes through the pushout. This achieves to prove that
$s_{*}k$ is finitely presented in $F_{*}\downarrow\mathcal{B}$. (Beware that
we need to consider a refinement relatively to $M$ and a refinement relatively
to $k$ separately before refining them jointly, as the sole common refinement
$(e,e^{\prime})$ relatively to $k$ may fail to equalize $F_{*}(a)$ and
$F_{*}(a^{\prime})$.)
∎

######  Lemma 3.3.

$\mathcal{G}_{F_{*}}$ form a dense generator.

###### Proof.

Let be $f:F_{*}(A)\rightarrow B$. Before anything, remark that the canonical
cone $A\simeq\mathcal{B}_{\omega}\downarrow A$ of $A$ is sent by $F_{*}$ to a
colimiting cone

$F_{*}(A)\simeq\underset{a:M\rightarrow A\atop
M\in\mathcal{A}_{\omega}}{{\textup{colim}}}F_{*}(M)$

This fact will be used twice in the following.

First, we must prove that the category consisting of all arrows of the form
$s_{*}k\rightarrow f$ is filtered in $F_{*}\downarrow\mathcal{B}$. Let be a
situation as below:

${{K_{1}}}$${{K_{1}^{\prime}}}$${{K_{2}}}$${{K_{2}^{\prime}}}$${{F_{*}(M_{1})}}$${{s_{1*}K_{1}^{\prime}}}$${{F_{*}(M_{2})}}$${{s_{2*}K_{2}^{\prime}}}$${{F_{*}(A)}}$${B}$$\scriptstyle{s_{1}}$$\scriptstyle{k_{1}}$$\scriptstyle{s_{1*}k_{1}}$$\scriptstyle{F_{*}(a_{1})}$$\scriptstyle{F_{*}(a_{2})}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$$\scriptstyle{k_{2}}$$\scriptstyle{s_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{f}$$\scriptstyle{s_{2*}k_{2}}$

Then $(F_{*}(a_{1})s_{1},b_{1}k_{1*}s_{1})$,
$(F_{*}(a_{2})s_{2},b_{2}k_{2*}s_{2})$, are part of the canonical cone of $f$
in $\mathcal{B}^{2}$, which is filtered, so that there exists some common
refinement as below

${{K_{1}}}$${{K^{\prime}_{1}}}$${{K_{2}}}$${{K_{2}^{\prime}}}$${K}$${{K^{\prime}}}$${{F_{*}(A)}}$${B}$$\scriptstyle{k_{1}}$$\scriptstyle{l_{1}}$$\scriptstyle{l_{2}}$$\scriptstyle{l_{1}^{\prime}}$$\scriptstyle{l_{2}^{\prime}}$$\scriptstyle{k_{2}}$$\scriptstyle{t}$$\scriptstyle{t^{\prime}}$$\scriptstyle{k}$$\scriptstyle{f}$

But from the filtered colimit decomposition of $F_{*}(A)$ above, we can find a
lift $s$ of $t$ through some $F_{*}(a):F_{*}(M)\rightarrow F_{*}(A)$ with $M$
finitely presented in $\mathcal{A}$, and moreover, this $M$ can be chosen as
equipped with a common refinement $u_{1}:M_{1}\rightarrow M$ of $a_{1}$ and
$u_{2}:M_{2}\rightarrow M$ of $a_{2}$ as below

${{K_{1}}}$${{K_{2}}}$${{F_{*}(M_{1})}}$${K}$${{F_{*}(M_{2})}}$${{F_{*}(M)}}$${{F_{*}(A)}}$$\scriptstyle{s_{1}}$$\scriptstyle{l_{1}}$$\scriptstyle{l_{2}}$$\scriptstyle{s_{2}}$$\scriptstyle{s}$$\scriptstyle{F_{*}(u_{1})}$$\scriptstyle{F_{*}(u_{2})}$$\scriptstyle{F_{*}(a_{1})}$$\scriptstyle{F_{*}(a_{2})}$

But as $l1^{\prime}$ and $l_{2}^{\prime}$ also are a common refinement of
$b_{1}$ and $b_{2}$, we end with a pair of morphisms $v_{1},v_{2}$ between the
induced pushouts as seen below

${{K_{1}^{\prime}}}$${{K_{2}^{\prime}}}$${{K_{1}}}$${{K_{2}}}$${{K^{\prime}}}$${K}$${{s_{1*}K_{1}^{\prime}}}$${{s_{2*}K_{2}^{\prime}}}$${{F_{*}(M_{1})}}$${{F_{*}(M_{2})}}$${{s_{*}K^{\prime}}}$${{F_{*}(M)}}$${B}$${{F_{*}(A)}}$$\scriptstyle{s_{1}}$$\scriptstyle{b_{1}}$$\scriptstyle{s}$$\scriptstyle{F_{*}(u_{1})}$$\scriptstyle{F_{*}(u_{2})}$$\scriptstyle{F_{*}(a_{1})}$$\scriptstyle{l_{1}^{\prime}}$$\scriptstyle{l_{2}^{\prime}}$$\scriptstyle{s_{*}k}$$\scriptstyle{s_{1*}K_{1}^{\prime}}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{2}}$$\scriptstyle{f}$$\scriptstyle{k_{1}}$$\scriptstyle{k_{2}}$$\scriptstyle{\lrcorner}$$\scriptstyle{s_{2*}K_{2}^{\prime}}$$\scriptstyle{k}$$\scriptstyle{s_{2}}$$\scriptstyle{l_{1}}$$\scriptstyle{l_{2}}$$\scriptstyle{b_{2}}$$\scriptstyle{F_{*}(a_{2})}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

which can be seen as a exhibiting $s_{*}k$ as equipped with a common
refinement $(u_{1},v_{1})$ and $(u_{2},v_{2})$ for $s_{1*}K_{1}$ and
$s_{2*}K_{2}$ respectively. Proving that a parallel pair between two such
arrows above $f$ also are equalized by a common refinement involves similar
arguments and can be left as an exercise.

Now we want to prove that $f$ is a filtered colimit of all the $s_{*}k$ above
it. First, observe that $f$ decomposes as a filtered colimit in
$\mathcal{B}^{2}$ $f\simeq{\textup{colim}}\,\mathcal{B}^{2}_{\omega}\downarrow
f$, which in particular induces that
$B\simeq{\textup{colim}}_{\mathcal{B}^{2}_{\omega}\downarrow f}{\rm cod}(k)$
for ${\rm cod}$ is finitary. But remember that $F_{*}$ also transported the
canonical cone of $A$ into a filtering colimit cocone, and as consequence for
each $(b,b^{\prime}):k\rightarrow f$ in $\mathcal{B}^{2}$ we have a
factorization as below

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{s_{*}K^{\prime}}}$${{F_{*}(A)}}$${B}$$\scriptstyle{f}$$\scriptstyle{k^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{s}$$\scriptstyle{\langle
F_{*}(a),b\rangle}$$\scriptstyle{F_{*}(a)}$$\scriptstyle{s_{*}k}$$\scriptstyle{b}$$\scriptstyle{\lrcorner}$

Moreover, if we denote by $I_{f}$ the category of all those factorizations
$(k,s,a,b,b^{\prime})$, which is obviously filtered from the previous items,
we have that $F_{*}(A)\simeq{\textup{colim}}_{(k,s,a,b,b^{\prime})\in
I_{f}}F_{*}({\rm dom}(a))$. From the previous observation, the following
projection is essentially surjective

$I_{f}\twoheadrightarrow\mathcal{B}_{\omega}^{f}\downarrow f$

Moreover, in each $a:M\rightarrow A$ we can define the category $I_{f,a}$
consisting of all the quintuplets of the form $(k,s,a,b,b^{\prime})$ with $a$
fixed; we have an inclusion $I_{f,a}\hookrightarrow I_{f}$, and a filtered
colimit

$F_{*}(M)=\underset{(k,s,a,b,b^{\prime})\in I_{f,a}}{{\textup{colim}}}\;K$

But now from lemma 2.11 we know that actually computing the colimit of all $k$
over $I_{f,a}$ is the same as computing the colimit of the corresponding
pushouts $s_{*}k$, that is we have an equality of filtered colimits in
$F_{*}(M)\downarrow\mathcal{B}$

$\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;k\simeq\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;s_{*}k$

and also, for the codomain functor preserves filtered colimits, we have a
filtered colimit in $\mathcal{B}$

$\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;K^{\prime}\simeq\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;s_{*}K^{\prime}$

But from the sequence of colimit decomposition

$F_{*}(A)\simeq\underset{\mathcal{A}_{\omega}\downarrow
A}{{\textup{colim}}}\,F_{*}(M)\simeq\underset{\mathcal{A}_{\omega}\downarrow
A}{{\textup{colim}}}\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;k$

we can use the isomorphism in each $a$ to get an isomorphism between the
following colimits:

$f\simeq\underset{\mathcal{A}_{\omega}\downarrow
A}{{\textup{colim}}}\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;k\simeq\underset{\mathcal{A}_{\omega}\downarrow
A}{{\textup{colim}}}\underset{(k,s,a,b,b^{\prime})\in
I_{f,a}}{{\textup{colim}}}\;s_{*}k$

This achieves to prove that the arrows of the form $s_{*}k$ are a generator of
finitely presented object in $F_{*}\downarrow\mathcal{B}$. For
$\mathcal{A}_{\omega}$ and $\mathcal{B}_{\omega}^{2}$ both are essentially
small while $\mathcal{B}$ is locally small, this category itself is
essentially small. This achieves to prove that $F_{*}\downarrow\mathcal{B}$ is
finitely accessible. And from what we said about existence of small limits, it
is moreover finitely accessible. ∎

From Finally, we would like to control explicitly the generator of finitely
presented objects of $F_{*}\downarrow\mathcal{B}$: we claim that not only
pushouts maps of $\mathcal{G}_{F_{*}}$ are a generator of etale objects, but
that any finitely presented object is actually of this form:

######  Lemma 3.4.

$\mathcal{G}_{F_{*}}$ is closed under retracts. Hence any finitely presented
object of $F_{*}\downarrow\mathcal{B}$ is in $\mathcal{G}_{F_{*}}$.

###### Proof.

Suppose that $f:F_{*}(A)\rightarrow B$ is finitely presented. Then from the
expression of $f$ as the filtered colimit
$f\simeq{\textup{colim}}\;{\mathcal{G}_{F_{*}}\downarrow f}$ established in
the previous item, we can exhibit $f$ as a retract of some arrow in
$\mathcal{G}_{F_{*}}$

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{s_{*}K^{\prime}}}$${{F_{*}(A)}}$${B}$${{F_{*}(A)}}$${B}$$\scriptstyle{F_{*}(s)}$$\scriptstyle{s}$$\scriptstyle{f}$$\scriptstyle{s^{\prime}}$$\scriptstyle{r^{\prime}}$$\scriptstyle{s_{*}k}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{F_{*}(r)}$$\scriptstyle{\lrcorner}$

But then from $\mathcal{A}_{\omega}$ is closed under retracts, $A$ must be in
$\mathcal{A}_{\omega}$. Moreover, this retraction can be transferred into a
retract of a finitely presented object in the coslice
$F_{*}(A)\downarrow\mathcal{B}$

${K}$${{K^{\prime}}}$${{F_{*}(M)}}$${{s_{*}K^{\prime}}}$${{F_{*}(A)}}$${{(F_{*}(r)s)_{*}K^{\prime}}}$${B}$${B}$$\scriptstyle{s_{*}k_{*}F_{*}(r)s^{\prime}}$$\scriptstyle{\langle
f,r^{\prime}\rangle}$$\scriptstyle{s_{*}k_{*}F_{*}(r)}$$\scriptstyle{k}$$\scriptstyle{s_{*}k}$$\scriptstyle{s}$$\scriptstyle{F_{*}(r)}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

But from lemma 2.10, we know this forces $f=\langle
f,r^{\prime}\rangle(F_{*}(r)s)_{*}k$ to be obtained as some pushout of a
finitely presented map: hence $f$ is actually in $\mathcal{G}_{F_{*}}$. ∎

Putting the previous lemma altogether we get the main result of this section:

######  Theorem 3.5.

For a morphism of locally finitely presentable categories
$F:\mathcal{A}\rightarrow\mathcal{B}$, the comma category
$F_{*}\downarrow\mathcal{B}$ is locally finitely presentable as well, and
moreover we have an equivalence

$(F_{*}\downarrow\mathcal{B})_{\omega}\simeq\mathcal{G}_{F_{*}}$

###### Proof.

From what precedes we know that $F_{*}\downarrow\mathcal{B}$ is finitely
accessible and $\mathcal{G}_{F_{*}}$ is exactly its generator of finitely
presented objects. Moreover, from what we saw at the begining of the section,
it has filtered colimits and small limits both computed in $\mathcal{B}^{2}$.
Therefore it is locally finitely presentable.
∎

######  Remark.

As a locally finitely presentable category, $F\downarrow\mathcal{B}$ also has
arbitrary colimits. However they are not computed in the arrow categories,
contrarily to filtered ones.

######  Proposition 3.6.

The codomain functor ${\rm
cod}:F_{*}\downarrow\mathcal{B}\rightarrow\mathcal{B}$ has a left adjoint,
which moreover sends finitely presented objects on finitely presented objects,
and the pair ${\rm cod}^{*}\dashv{\rm cod}$ defines a morphism of locally
finitely presentable categories.

###### Proof.

Observe first that ${\rm cod}$ factorizes through the inclusion into the arrow
category

${{F_{*}\downarrow\mathcal{B}}}$${{\mathcal{B}}}$${{\mathcal{B}^{2}}}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{\iota_{F}}$$\scriptstyle{{\rm cod}}$$\scriptstyle{\simeq}$

Then ${\rm cod}:\mathcal{B}^{2}\rightarrow\mathcal{B}$ has as left adjoint the
functor $!_{(-)}$ sending $B$ on the initial map $!_{B}:0\rightarrow B$, and
any $f:B_{1}\rightarrow B_{2}$ on the induced square

${0}$${{B_{1}}}$${0}$${{B_{2}}}$$\scriptstyle{!_{B_{1}}}$$\scriptstyle{f}$$\scriptstyle{!_{B_{2}}}$

Indeed, any square $!_{B}\rightarrow f$ for $f:{\rm dom}(f)\rightarrow{\rm
cod}(f)$ as below

${0}$${{{\rm dom}(f)}}$${B}$${{{\rm
cod}(f)}}$$\scriptstyle{!_{B}}$$\scriptstyle{!_{B_{1}}}$$\scriptstyle{f}$$\scriptstyle{g}$

is uniquely defined by a choice of $g:B\rightarrow{\rm cod}(f)$, which proves
that $!_{(-)}\dashv{\rm cod}$.

Now, we construct a left adjoint $\iota_{F}^{*}$ of
$\iota_{F}:F_{*}\downarrow\mathcal{B}\hookrightarrow\mathcal{B}^{2}$ as
follows: for any arrow $f:B_{1}\rightarrow B_{2}$ takes the pushout along the
unit of $B_{1}$

${{B_{1}}}$${{B_{2}}}$${{F_{*}F^{*}(B_{1})}}$${{\eta_{B_{1}*}B_{2}}}$$\scriptstyle{\eta_{B_{1}}}$$\scriptstyle{f}$$\scriptstyle{\eta_{B_{1}*}f}$$\scriptstyle{\lrcorner}$

and for a morphism of arrows $(u,v):f\rightarrow f^{\prime}$, take as
$\iota^{*}_{F}(u,v)$ the vertical arrow of the front square in the diagram
below

${{B_{1}}}$${{B_{2}}}$${{F_{*}F^{*}(B_{1})}}$${{\eta_{B_{1}*}B_{2}}}$${{B_{1}^{\prime}}}$${{B_{2}^{\prime}}}$${{F_{*}F^{*}(B^{\prime}_{1})}}$${{\eta_{B^{\prime}_{1}*}B^{\prime}_{2}}}$$\scriptstyle{\eta_{B_{1}}}$$\scriptstyle{f}$$\scriptstyle{u}$$\scriptstyle{v}$$\scriptstyle{\eta_{B_{1}*}f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\eta_{B^{\prime}_{1}*}f^{\prime}}$$\scriptstyle{\eta_{B^{\prime}_{1}}}$$\scriptstyle{F_{*}F^{*}(u)}$$\scriptstyle{\langle\eta_{B^{\prime}_{1}*}f^{\prime}F_{*}F^{*}(u),f^{\prime}_{*}\eta_{B^{\prime}_{1}}v\rangle}$$\scriptstyle{\lrcorner}$

Then for any morphism of arrow in $\mathcal{B}^{2}$ as below

${{B_{1}}}$${{B_{2}}}$${{F_{*}(A)}}$${B}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{u}$$\scriptstyle{v}$

the adjunction $F^{*}\dashv F_{*}$ returns a unique morphism
$u:F^{*}(B_{1})\rightarrow A$ in $\mathcal{A}$ such that $u$ factorizes
uniquely as $F_{*}(\overline{u})\eta_{B_{1}}$, and then this induces uniquely
a morphism of arrows as seen below in the front square

${{B_{1}}}$${{B_{2}}}$${{F_{*}F^{*}(B_{1})}}$${{\eta_{B_{1}*}B_{2}}}$${{F_{*}(A)}}$${B}$$\scriptstyle{\eta_{B_{1}}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\overline{u}}$$\scriptstyle{\langle
g\overline{u},v\rangle}$$\scriptstyle{u}$$\scriptstyle{v}$$\scriptstyle{\lrcorner}$$\scriptstyle{\eta_{B_{1}*}f}$

Now we want to compose those functors to get a left adjoint ${\rm
cod}^{*}:\mathcal{B}\rightarrow\mathcal{B}^{2}$. First observe that
$F^{*}(0)\simeq 0$ as $F^{*}$ preserves colimits; then the unit
$\eta_{0}:0\rightarrow F_{*}F^{*}(0)$ coincides with the initial map
$!_{F_{*}(0)}:0\rightarrow F_{*}(0)$. Now take a $B$ in $\mathcal{B}$ to the
lower arrow $\eta_{0}*!_{B}$ in the following pushout

${0}$${B}$${{F_{*}(0)}}$${{F_{*}(0)+B}}$$\scriptstyle{!_{B}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\eta_{0}*!_{B}}$$\scriptstyle{\lrcorner}$

This defines a left adjoint to ${\rm cod}$: indeed, though $F_{*}$ does not
preserve initialness, initialness is remembered in some sense in the comma any
morphsim $\eta_{B*}f\rightarrow g$ with $g:F_{*}(A)\rightarrow B^{\prime}$ has
is left component forced to be $F*(!_{A})$, so that in the diagram below

${0}$${B}$${{F_{*}(0)}}$${{F_{*}(0)+B}}$${{F_{*}(A)}}$${{B^{\prime}}}$$\scriptstyle{\eta_{0}*!_{B}}$$\scriptstyle{!_{B}}$$\scriptstyle{\eta_{0}}$$\scriptstyle{F^{*}(!_{A})}$$\scriptstyle{v}$$\scriptstyle{g}$$\scriptstyle{\lrcorner}$

we see that such an arrow is uniquely determined by a map $v!_{B*}\eta_{0}$,
which itself was in return uniquely determined by $v$. Hence the adjunction.

Now, though it is also a consequence of ${\rm cod}$ being finitary, it is easy
to see directly that this left adjoint preserves finite presentedness with our
notion of finitely presented objects in $F_{*}\downarrow\mathcal{B}$. Indeed
if $K$ is in $\mathcal{B}_{\omega}$, then not only is the initial map
$!_{K}:0\rightarrow K$ in $\mathcal{B}_{\omega}$ as 0 always is finitely
presented, but as $F^{*}(0)=0$ is in $\mathcal{A}_{\omega}$, then ${\rm
cod}^{*}(B)=\eta_{0*}!_{B}$ is finitely presented in
$F_{*}\downarrow\mathcal{B}$ from what was proved above. ∎

Now we turn to the other part of the factorization:

######  Proposition 3.7.

The functor $1_{F_{*}}:\mathcal{A}\rightarrow F_{*}\downarrow\mathcal{B}$ has
a left adjoint $1^{*}_{F_{*}}$, which moreover sends finitely presented
objects on finitely presented objects, and the pair $1^{*}_{F_{*}}\dashv
1_{F_{*}}$ defines a morphism of locally finitely presentable categories.

###### Proof.

We saw that $1_{F_{*}}$ is continuous and accessible by the very computation
of limits in the comma. Applying the adjoint functor theorem would ensure the
existence of a left adjoint: however we want an explicit description of this
left adjoint. From adjoint functor theorem, we can compute its value at a
given object as a limit as follows.

Being finitary, $1_{F_{*}}$ is accessible, hence satisfies the solution set
condition: thus for each $f:F_{*}(A)\rightarrow B$, the comma category
$f\downarrow 1_{F_{*}}$ has a small weakly initial family $J_{f}$, whose
elements will be denoted as

${{F_{*}(A)}}$${B}$${{F_{*}(A_{i})}}$${{F_{*}(A_{i})}}$$\scriptstyle{F_{*}(a_{i})}$$\scriptstyle{f}$$\scriptstyle{b_{i}}$

for each $i\in J_{f}$. Then define the value of $1^{*}_{F_{*}}$ as the limit

$1^{*}_{F_{*}}(f)=\underset{j\in J_{f}}{\lim}\;A_{i}$

Then it is standard calculation to see that this defines a left adjoint to
$1_{F_{*}}$. Moreover, the fact it restricts to finitely presented objects is
a consequence of $1_{F_{*}}$ being finitary.
∎

Uniqueness of adjoints ensures that the composite of those left adjoints
coincides up to invertible 2-cell with the left adjoint $F^{*}$ as depicted in
the factorization below:

${{\mathcal{A}_{\omega}}}$${{\mathcal{B}_{\omega}}}$${{(F_{*}\downarrow\mathcal{B})_{\omega}}}$$\scriptstyle{F^{*}_{\omega}}$$\scriptstyle{{\rm
cod}^{*}_{\omega}}$$\scriptstyle{(1_{F_{*}})_{\omega}^{*}}$$\scriptstyle{\simeq}$

However we would like to understand explicitely what makes $1^{*}_{F_{*}}$ to
return the finitely presented object $F^{*}(K)$ when applied to ${\rm
cod}^{*}(K)$ for a finite presented $K$, the precedent proof failing to
provide a satisfactory description of the process.

Recall that we constructed ${\rm cod}^{*}(K)$ as
$\eta_{0*}!_{K}:F_{*}(0)\rightarrow F_{*}(0)+K$. Then observe that the
following square

${0}$${K}$${{F_{*}(0)}}$${{F_{*}F^{*}(K)}}$${{F_{*}F^{*}(K)}}$$\scriptstyle{F_{*}(!_{K})}$$\scriptstyle{\eta_{0}}$$\scriptstyle{!_{K}}$$\scriptstyle{\eta_{K}}$

induces a unique factorization

${0}$${K}$${{F_{*}(0)}}$${{F_{*}(0)+K}}$${{F_{*}F^{*}(K)}}$${{F_{*}F^{*}(K)}}$$\scriptstyle{F_{*}(!_{K})}$$\scriptstyle{\eta_{0}}$$\scriptstyle{!_{K}}$$\scriptstyle{\eta_{K}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\langle
F_{*}(!_{K}),\eta_{K}\rangle}$

Then it is clear that the following square

${{F_{*}(0)}}$${{F_{*}(0)+K}}$${{F_{*}F^{*}(K)}}$${{F_{*}F^{*}(K)}}$$\scriptstyle{F_{*}(!_{K})}$$\scriptstyle{{\rm
cod}^{*}(K)}$$\scriptstyle{\langle F_{*}(!_{K}),\eta_{K}\rangle}$

is an initial object of the comma ${\rm cod}^{*}(K)\downarrow 1_{F_{*}}$, and
in particular factorizes all the members of the solution set of $1_{F_{*}}$ at
${\rm cod}^{*}(F)$: hence the limit above reduces here on

$1_{F_{*}}^{*}({\rm cod}^{*}(K))=\underset{{\rm cod}^{*}(K)\downarrow
1_{F_{*}}}{\lim}\;A\simeq F^{*}(K)$

Observe also that the pseudosection $\pi_{1}$ of $1_{F_{*}}$ is part of a
morphism of locally finitely presentable categories: in fact, it is immediate
to see that it is _right adjoint_ to $\pi_{1}$, and that its both finitary and
continuous. To sum up, the following square

${{F_{*}\downarrow\mathcal{B}}}$${{\mathcal{B}}}$${{\mathcal{A}}}$${{\mathcal{B}}}$$\scriptstyle{F}$$\scriptstyle{\pi_{1}}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{\lambda_{F_{*}}}$

actually lies inside of ${\bf LFP}$.

######  Theorem 3.8.

For $F:\mathcal{A}\rightarrow\mathcal{B}$ in ${\bf LFP}$,
$F_{*}\downarrow\mathcal{B}$ together with ${\rm cod}$ and $\pi_{1}$ is the
comma object $F\downarrow\mathcal{B}$ in ${\bf LFP}$.

###### Proof.

Any other 2-cell in ${\bf LFP}$

${C}$${{\mathcal{B}}}$${{\mathcal{A}}}$${{\mathcal{B}}}$$\scriptstyle{F}$$\scriptstyle{G}$$\scriptstyle{H}$$\scriptstyle{\lambda}$

defines from its underlying 2-cell in ${\bf Cat}$ a unique arrow $S_{\lambda}$
as below

${C}$${{F_{*}\downarrow\mathcal{B}}}$${{\mathcal{B}}}$${{\mathcal{A}}}$${{\mathcal{B}}}$$\scriptstyle{F_{*}}$$\scriptstyle{G_{*}}$$\scriptstyle{H_{*}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{{\rm
cod}}$$\scriptstyle{S_{\lambda}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\lambda_{F_{*}}}$

sending any $C$ in $\mathcal{C}$ to the corresponding component of the natural
transformation $\lambda$

${{F_{*}G_{*}(C)}}$${{H_{*}(C)}}$$\scriptstyle{\lambda_{C}}$

But from $F_{*}$, $G_{*}$ and $H_{*}$ all are continuous and finitary, and
from the computation of limits and filtered colimits in the comma,
$S_{\lambda}$ is itself continuous and finitary. Moreover, from uniqueness of
adjoints, any two 2-cells inducing the same functor in ${\bf LFP}$ induce in
fact a same factorization in ${\bf Cat}$ and must then be equivalent.

∎

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