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In category theory, a branch of mathematics, a dinatural transformation α {\displaystyle \alpha } between two functors
written
is a function that to every object c {\displaystyle c} of C {\displaystyle C} associates an arrow
and satisfies the following coherence property: for every morphism f : c → c ′ {\displaystyle f:c\to c'} of C {\displaystyle C} the diagram
commutes.[1] Note the direction of S ( f , g ) {\displaystyle S(f,g)} is opposite along f {\displaystyle f} in the first component since it is contravariant.
The composition of two dinatural transformations need not be dinatural.
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