Dinatural transformation

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

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

written

${\displaystyle \alpha :S{\ddot {\to }}T,}$

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

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

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

commutes.^[1] Note the direction of ${\displaystyle S(f,g)}$ is opposite along ${\displaystyle f}$ in the first component since it is contravariant.

The composition of two dinatural transformations need not be dinatural.