For $C$ a small category, its category of presheaves is the functor category
from the opposite category of $C$ to Set.
An object in this category is a presheaf. See there for more details.
The category of presheaves $PSh(C)$ is the free cocompletion of $C$.
the Yoneda lemma says that the Yoneda embedding $j : C \to PSh(C)$ is – in particular – a full and faithful functor.
A category of presheaves is a topos.
The construction of forming (co)-presheaves extends to a 2-functor
from the 2-category Cat to the 2-category Topos. (See at geometric morphism the section Between presheaf toposes for details).
A reflective subcategory of a category of presheaves is a locally presentable category if it is closed under $\kappa$-directed colimits for some regular cardinal $\kappa$ (the embedding is an accessible functor).
A sub-topos of a category of presheaves is a Grothendieck topos: a category of sheaves (see there for details).
See functoriality of categories of presheaves.
A category $E$ is equivalent to a presheaf topos if and only if it is cocomplete, atomic, and regular.
This is due to Marta Bunge.
As every topos, a category of presheaves is a cartesian closed monoidal category.
For details on the closed structure see
Let $C$ be a category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$. Write $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$ and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the Yoneda embedding.
There is an equivalence of categories
The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.
A weak inverse of $e$ is given by the functor
which sends $\eta : F' \to Y(C))$ to $F \in PSh(C/c)$ given by
where $F'(d)|_c$ is the pullback
Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $C$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:
Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the closed monoidal structure on presheaves.
See also functors and comma categories.
For the analog statement in (∞,1)-category theory see
Consider $\int_C Y(c)$ , the category of elements of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.
Hence, the above proposition 1 can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:
Let $P:C^{op}\to Set$ be a presheaf. Then there is an equivalence of categories
For a proof see Kashiwara-Schapira (2006, p.26). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).
In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.
A finite presheaf on a category $C$ is a functor $C^{op}\to FinSet$ valued in the category of finite sets. Categories of finite presheaves will hardly be Grothendieck toposes for want of infinite limits but they still can turn out to be elementary toposes as e.g. in the case of $FinSet$ itself.
By going through the proof that ordinary categories of presheaves are toposes, one observes that the constructions stay within finite presheaves when applied to a finite category $C$ i.e. one with only a finite set of morphisms. Hence, one has the following
Let $C$ a finite category. Then the category of finite presheaves $[C^{op},FinSet]$ is a topos. $\qed$
Note, that the category $[G,FinSet]$ of finite $G$-sets is topos even when the group $G$ is infinite! In this case it is crucial that $\Omega =\{\emptyset , G\}$ in $[G,Set]$ is a finite set.
(Cf. Borceux (1994, p.299))
See at models in presheaf toposes.
For (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
A classical (advanced) reference is exposé 1 of
An elementary introduction to presheaf toposes emphasizing finite underlying categories $C$ is
Standard references are
Francis Borceux, Handbook of Categorical Algebra 3 : Categories of Sheaves , Cambridge UP 1994.
Masaki Kashiwara, Pierre Schapira, Categories and Sheaves , Springer Heidelberg 2006.
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.