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 \( \mathcal{E} \).
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For , a dinatural transformation has components satisfying, for ,Both composites have domain and codomain . It is the compatibility used for categorical ends and categorical coends, rather than ordinary naturality on just one variance.