Dinatural transformation

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

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

written

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

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

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

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

commutes.

The composition of two dinatural transformations need not be dinatural.