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 $s\in \mathcal{S}$.

1The 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.

2For 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).

3Suppose 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.