Dinatural Transformation

Definition

Let $\mathcal{C},\mathcal{D}$ be categories and let $F,G:{\mathcal{C}}^{op}\times \mathcal{C}\to \mathcal{D}$ be bifunctors.
A dinatural transformation $\alpha :F\stackrel{..}{\to }G$ consists of a family of morphisms in $\mathcal{D}$

${\alpha }_c:F(c,c)\to G(c,c)\forall c:\mathcal{C}$

such that for every morphism $f:c\to c^{\prime }$ in $\mathcal{C}$ the following hexagon commutes:

$G({\mathrm{id}}_c,f)\circ {\alpha }_c\circ F(f,{\mathrm{id}}_c)=G(f,{\mathrm{id}}_{c^{\prime }})\circ {\alpha }_{c^{\prime }}\circ F({\mathrm{id}}_{c^{\prime }},f)$

Remarks

A dinatural transformation is weaker than a natural transformation between $F$ and $G$: a natural transformation requires a morphism $F(c,c^{\prime })\to G(c,c^{\prime })$ for every pair $(c,c^{\prime })$, whereas dinaturality is only required on the diagonal.

Dinatural transformations arise in the definition of ends and coends, where the universal wedge is a dinatural transformation from a constant functor.

See also

End (Category Theory)
Coend (Category Theory)
Profunctor

References

gylterud2011-symmetric-containers-thesis
https://ncatlab.org/nlab/show/dinatural+transformation