Dinatural transformation

In category theory, a branch of mathematics, a dinatural transformation $$\alpha$$ between two functors


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

written


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

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


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

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.