Dinatural transformation

In category theory, a dinatural transformation $α$ between two functors

$S,T : \mathrm{C}^{\mathrm{op}}\times\mathrm{C}\to\mathrm{X},$

written

$\alpha : S\ddot\to T,$

is a function which to every object c of C associates an arrow

$\alpha_c : S(c,c)\to T(c,c)$ of X

and satisfies the following coherence property: for every morphism $f:c\to c'$ of C the diagram

commutes.

The composition of two dinatural transformations need not be dinatural.

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