Functor Categories as Topoi

Consider the category of functors between two categories $E:=D^C$, with D being Sets.  It may be shown that E is complete and co-complete due to the fact Sets is; thinking of each object in E as a map from C to Sets, in the sense of evaluation, it can be shown that $(\varinjlim F_i) (C)= \varinjlim F_i(C)$ where $F_i$ are objects in E.  In particular $-\times E: E \rightarrow E$ commultes with both types of limits.  

Essentially E inheriting Sets' properties, similarly it can be shown that it is abelian, with enough injective objects (again see See Tamme).   

On objects A of E, PE(A) defined to be the set of subfunctors $Hom_E(-,A)xE$ is a functor with values in E, and the subfunctors FxE are in bijection with the morphisms in Nat(F,PE).  Hence E is a topos.  

The Functor category E as described above is called a presheaf, and in the case of a locally small category, via a Yoneda Embedding it may be though of as enlarging an arbitrary category C in such a way, that the enriched C will be complete, co-complete, abelian and posses enough injectives.  

A little something on Topoi

A Topos E is a mathematical object called a locally small Category, which has limits and a power object.  That is to say that for any object A of E the functor Sub($- \times A$) of objects which factor through $- \times A$ is  non-empty member of the category $Sets$.  Moreover there is an object of E called the power object of A for which Hom(-,PA)$\cong$  Sub(-,-$\times$ A).

Note: recall since E is locally small Hom(-PA) is a set so the equivalence of objets above is a bijection in Sets.

Some examples of Topoi would be the category sets; here for any object X PX is $\mathscr{P}$(X) its power set.  With the natural equivalence of $\phi$: Hom(-,PA)$\cong$  Sub(-,-$\times$ A) to be $\phi$:Hom(1,PB)$\rightarrow$Sub(B).