 Dinatural transformation

For $F,G:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$, a dinatural transformation has components ${\alpha }_C:F(C,C)\to G(C,C)$ satisfying, for $f:C\to D$,Both composites have domain $F(D,C)$ and codomain $G(C,D)$. It is the compatibility used for categorical ends and categorical coends, rather than ordinary naturality on just one variance.

A dinatural transformation is a concept in category theory, specifically in the context of functors and natural transformations. It generalizes the notion of a natural transformation to situations involving two different functors that are indexed by a third category. In more detail, consider two categories \( \mathcal{C} \) and \( \mathcal{D} \), along with a third category \( \mathcal{E} \).