In category theory, a branch of mathematics, a dinatural transformation α {\displaystyle \alpha } between two functors

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 coherence property: 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.