Editing Functor category (section)

== Facts ==
Most constructions that can be carried out in <math>D</math> can also be carried out in <math>D^C</math> by performing them "componentwise", separately for 
each object in <math>C</math>. For instance, if any two objects <math>X</math> and <math>Y</math> in <math>D</math> have a [[product (category theory)|product]] <math>X\times Y</math>, 
then any two functors <math>F</math> and <math>G</math> in <math>D^C</math> have a product <math>F\times G</math>, defined by <math>(F \times G)(c) = F(c)\times G(c)</math> 
for every object <math>c</math> in <math>C</math>. Similarly, if <math>\eta_c : F(c) \to G(c)</math> is a natural transformation and each <math>\eta_c</math> has a kernel <math>K_c</math> in the category <math>D</math>, 
then the kernel of <math>\eta</math> in the functor category <math>D^C</math> is the functor <math>K</math> with <math>K(c) = K_c</math> for every object <math>c</math> in <math>C</math>.

As a consequence we have the general [[rule of thumb]] that the functor category <math>D^C</math> shares most of the "nice" properties of <math>D</math>:
* if <math>D</math> is [[complete category|complete]] (or cocomplete), then so is <math>D^C</math>;
* if <math>D</math> is an [[abelian category]], then so is <math>D^C</math>;

We also have:
* if <math>C</math> is any small category, then the category <math>\textbf{Set}^C</math> of [[presheaf (category theory)|presheaves]] is a [[topos]].

So from the above examples, we can conclude right away that the categories of directed graphs, <math>G</math>-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of <math>G</math>, modules over the ring <math>R</math>, 
and presheaves of abelian groups on a topological space <math>X</math> are all abelian, complete and cocomplete.

The embedding of the category <math>C</math> in a functor category that was mentioned earlier uses the [[Yoneda lemma]] as its main tool. For every object <math>X</math> of <math>C</math>, 
let <math>\text{Hom}(-,X)</math> be the contravariant [[representable functor]] from <math>C</math> to <math>\textbf{Set}</math>. The Yoneda lemma states that the assignment 
:<math>X \mapsto \operatorname{Hom}(-,X)</math>
is a [[full embedding]] of the category <math>C</math> into the category Funct(<math>C^\text{op}</math>,<math>\textbf{Set}</math>). So <math>C</math> naturally sits inside a topos.

The same can be carried out for any preadditive category <math>C</math>: Yoneda then yields a full embedding of <math>C</math> into the functor category Add(<math>C^\text{op}</math>,<math>\textbf{Ab}</math>). 
So <math>C</math> naturally sits inside an abelian category.

The intuition mentioned above (that constructions that can be carried out in <math>D</math> can be "lifted" to <math>D^C</math>) can be made precise in several ways; the most succinct formulation uses the language of [[adjoint functors]]. 
Every functor <math>F : D \to E</math> induces a functor <math>F^C : D^C \to E^C</math> (by composition with <math>F</math>). If <math>F</math> and <math>G</math> is a pair of adjoint functors, then <math>F^C</math> and <math>G^C</math> is also a pair of adjoint functors.

The functor category <math>D^C</math> has all the formal properties of an [[exponential object]]; in particular the functors from <math>E \times C \to D</math> 
stand in a natural one-to-one correspondence with the functors from <math>E</math> to <math>D^C</math>. The category <math>\textbf{Cat}</math> of all small categories with functors as morphisms 
is therefore a [[cartesian closed category]].