Dinatural transformation

In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a dinatural transformation 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 between two <a href="/facts/Functor/l2u3rIBE">functors</a>

 
 
 
 S
 ,
 T
 :
 
 C
 
 
 o
 p
 
 
 
 ×
 C
 →
 D
 ,
 
 
 {\displaystyle S,T:C^{\mathrm {op} }\times C\to D,}

written

α
        :
        S
        
          
            
              →
              ¨
            
          
        
        T
        ,
      
    
    {\displaystyle \alpha :S{\ddot {\to }}T,}

is a function that to every object 
 
 
 
 c
 
 
 {\displaystyle c}
 
 of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 associates an arrow

α
          
            c
          
        
        :
        S
        (
        c
        ,
        c
        )
        →
        T
        (
        c
        ,
        c
        )
      
    
    {\displaystyle \alpha _{c}:S(c,c)\to T(c,c)}
  
 of 
  
    
      
        D
      
    
    {\displaystyle D}

and satisfies the following <a href="/facts/Coherence_property/KPbATej5">coherence property</a>: for every morphism 
 
 
 
 f
 :
 c
 →
 
 c
 ′
 
 
 
 {\displaystyle f:c\to c'}
 
 of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 the diagram

commutes.
The composition of two dinatural transformations need not be dinatural.

Dinatural transformation open-in-new

Dinatural transformation