Functor category

<p>In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a functor category 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 is a category where the objects are the <a href="/facts/Functor/l2u3rIBE">functors</a> 
  
    
      
        F
        :
        C
        →
        D
      
    
    {\displaystyle F:C\to D}
  
 and the <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> are <a href="/facts/Natural_transformation/xwPumTHt">natural transformations</a> 
  
    
      
        η
        :
        F
        →
        G
      
    
    {\displaystyle \eta :F\to G}
  
 between the functors (here, 
  
    
      
        G
        :
        C
        →
        D
      
    
    {\displaystyle G:C\to D}
  
 is another object in the category). Functor categories are of interest for two main reasons: 
</p>
<ul><li>many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;</li>
<li>every category embeds in a functor category (via the <a href="/facts/Yoneda_embedding/j04vXd6x">Yoneda embedding</a>); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.</li></ul>

Functor category open-in-new

Functor category