Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$. 


Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\text{Hom}\_{\text{Psh}\left(\mathcal{C}\right)}(-,X)=h\_X$ is such that $h\_X(s)$ is an isomorphism for every $s\in \mathcal{S}$.



> 
> **1** The full subcategory of $\mathcal{S}$-sheaves, call it $P\_{\mathcal S}$, comes equipped with a localization functor $L\_{\mathcal S}$ which is left adjoint to the inclusion, so that $P\_{\mathcal S}\subset \text{Psh}\left(\mathcal{C}\right)$ is a reflective subcategory. 
> 
> 
> **2** For every co-complete category $\mathcal{E}$ and for every co-continuous functor $F:\mathcal{C}\to \mathcal{E}$ with the property that $\text{Lan}\_yF(s)$ is invertible for every $s \in \mathcal S$, there exists a unique co-continous functor $\overline F:P\_{\mathcal S}\to \mathcal {E}$ such that $\overline F L\_{\mathcal S}y=F$ (where $y$ is the Yoneda embedding).
> 
> 
> **3** Suppose that $\mathcal S$ is a family of subfunctors of representable functors and $\mathcal S$ is a Grothendieck topology.
>  Then in this case $P\_{\mathcal S}$ is a topos, and every (Grothendieck) topos arises in this way for some choice of $\mathcal S$
> 
> 
> 


I would like to know a reference for **1**,**2**,**3**. Is there a book where the notion of topos is defined in this way? 


If you don't know a reference but can sketch me an idea of the proofs I would still be happy.



Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos
1. (&3?) Can follow from the theory of orthogonality classes. A reference is **Thm 1.38** and **1.39** in **Locally presentable and accessible categories** by *Adamek and Rosicky.*
2. I do not have a reference, but this is really a standard technique. In my mind, this follows from the theory of Kan extensions. A place where this kind of arguments is carried on is in my preprint with *Julia Ramos Gonzales*, **Gabriel-Ulmer duality for topoi and its relation with site presentations**, **Prop 4.3.** The latter is far from being a natural reference, and probably someone can suggest better. In the spirit of this question, I should probably mention **Cor VII.7.4** in **Sheaves in Geometry and Logic** by *Mac Lane and Moerdijk*.
3. Choose your favourite generator $S$ of your topos $i: S \subset\mathcal{E}$, and use the *nerve* of $i$, $N(i): \mathcal{E} \to \mathsf{Set}^{S^\circ}$. Since in a topos every generator is dense, the nerve is fully faithful, moreover, it preserves limits and by the general AFT (here any argument would work, you can also use the universal property of the presheaf construction), it has a left adjoint. You can check (it's easier if $S$ is closed under finite limits) that the left adjoint must preserve finite limits (because $\mathcal{E}$ is a topos) . This kind of *representation* argument will work for every cocomplete category with a dense generator and sits at the very core of **Thm. 1.46** in **LPAC**.


