**About the Ind completions of a category $\mathcal{C}$ :**

Let $\mathcal{C}^>:=Fun(\mathcal{C}^{op}\to Set)$ the presheaves category.

we have the yoneda full embedding $Y: \mathcal{C}\to \mathcal{C}^>: X\mapsto h_X$.

For $P\in \mathcal{C}^>$ we indicate by $\int P$ the the comma category $\mathcal{C}\downarrow P$, with objects the couples $(X,x)$ with $X\in \mathcal{C},\ x\in P(X)$ and morphisms $f: (X, x)\to (Y, y)$ the $f\in \mathcal{C}(X, Y)$ such that $P(f)(y)=x$, we have the natural functor $\pi_P: \int P\to \mathcal{C}: (X, x)\mapsto X$ and is a foundamental theorem that $P\cong colim_{(X, x)\in \int P}\ h_X= colim\ Y\circ \pi $

we call $P$ **flat** if there is a final functor $\phi: I\to \int P$ (i.e. for any $\xi\in \int P$ the comma category $\xi\downarrow \phi$ is nonempty and connected) with $I$ a small filtered category (i.e. for any $\forall i, j\in I: \exists k\in I:\ I(i, k)\cup I(j, k)\neq\emptyset$ and for $\phi, \psi: I\to j$ exist $\theta: j\to k$ with $\theta\circ\phi=\theta\circ\psi$).

A flat functor $P$ preserve finite limit:

we can write $P\cong colim_{i\in I} h_{X_i}$ then

$P(Y)=colim_{i\in I}\mathcal{C}(Y, X_i)$ then if $Y= colim_{j\in J}$ then

$lim_{j\in J}P(Y_j)\cong
lim_{j\in J}\ colim_{i\in I}\mathcal{C}(Y_j, X_i)\cong$
$\ colim_{i\in I}\ lim_{j\in J}\mathcal{C}(Y_j, X_i)\cong $
$colim_{i\in I}\mathcal{C}(colim_{j\in J} Y_j, X_i)\cong $
$colim_{i\in I}\mathcal{C}(Y, X_i)\cong P(Y)$.

Let $P_1(\mathcal{C})\subset\mathcal{C}^>$ the full subcategory of the flat presheaves. The category $P_1(\mathcal{C})$ has the small filtrant colimits (no too easy to prove).

Of course any representable is flat, then we have the Yoneda immersion $Y: \mathcal{C}\to P_1(\mathcal{C})$
A flat presheaf $P$ preserve finite colimits:

let $J$ a finite diagram, for any $P\in P_1(\mathcal{C})$ we have:

$(Y(colim_{j\in J}Y_j), P)\cong P(colim_{j\in J}Y_j)\cong$
$ lim_{j\in J}P(Y_j)\cong lim_{j\in J}(h_{Y_j}, P)\cong $
$(colim_{j\in J} h_{Y_j}, P)$

from Yoneda lemma follow that $colim_{j\in J}Y_j\cong colim_{j\in J} h_{Y_j}$ in $P_1(\mathcal{C})$.

We have the following universal property: let $F: \mathcal{C}\to \mathcal{A}$ where $\mathcal{A}$ has the filtered little colimits, then exist unique (to isomorphisms) the $F_1: P_1(\mathcal{C})\to \mathcal{A} $ preserving the small filtered colimits, with $F=F_1\circ Y$ (let $F_1:=Lan_{Y} F$ the left Kan extension of $F$ respect to $Y$).

Now, gived a $P\in P_1(\mathcal{C})$ we represent it as a filtered system $(X_i)_{i\in I}$

(with $P\cong colim_{i\in I}h_{X_i}$)

how we could represent a morfisms $\Phi: P\to Q$ (let fixed a representation $(Y_j)_{j\in J}$ of $Q$) ?

for $i\in I$ considering $h_{X_i}\to P\to Q\cong colim_{j\in J}h_{Y_j}$

by yoneda lemma, and the definition of colimit in $Set$, this has a factorization on some $f(i)\in J$ i.e. like some $\Phi_i: h_{X_i}\to h_{Y_{f(i)}}\to Q$

Then choosing for any $i\in I$ as above we have a map $f: I\to J$ and a famili o morphisms $(\Phi_i: X_i\to Y_{f(i)})_{i\in I}$, with the following property:

gived a $\phi: i\to i'$ we have a $k\in J$ and two morphisms $\psi: \Phi(i)\to k,\ \psi': \Phi(i')\to k$ such that
$\psi'\circ\Phi_{i'}\circ X_\phi= \psi \circ \Phi_i$ (where $X_\phi: X_i\to X_{i'}$ is the canonical morphism).

Vice versa such data detecting a unique morphisms $\Phi: P\to Q$ that has such data as representation as above.

If $\Phi: P\to Q$ and $\Psi: Q\to R$ have the representation $(f, \phi)$ and $(g, \psi)$ (and $R\cong colim_{k\in K}h_{Z_k}$ then the composition $\Psi\circ \Phi$ is $(g\circ f, \psi\ast_{f, g} \phi)$ where $(\psi\ast_{f, g} \phi)_i: X_i\to Y_{f(i)}\to Z_{gf(i)}$.

The idetity $1_P: P\to P$ as the representation $(1_I, (1_: i\to i)_{i\in I}$.

Then we get a new category $Ind(\mathcal{C})$ equivalent to $P_1(\mathcal{C})$ with object the filtered diagrams in $\mathcal{C}$ like $(X_i)_{i\in I}$ and for morphism the representation $(f, \Phi)$ as above, with the above compositions and identities.

For the universal property above, follow a natural equivalence $P_1(P_1(\mathcal{C}))\cong P_1(\mathcal{C})$ then we have a natural equivalence $Ind(Ind(\mathcal{C}))\cong Ind(\mathcal{C})$.