nLab strong dinatural transformation


A strong dinatural transformation is a notion of natural transformation between two functors C op×CDC^{op}\times C\to D that is stronger than a dinatural transformation.

Unlike dinatural transformations, strong dinatural transformations can always be composed. They have close connections to parametricity in computer science.


Let F,G:C op×CDF,G:C^{op}\times C\to D be functors. A strong dinatural transformation α:FG\alpha :F\to G consists of, for each cCc\in C, a component α c:F(c,c)G(c,c)\alpha_c : F(c,c) \to G(c,c), such that for any morphism f:ccf:c\to c' in CC and any span F(c,c)WF(c,c)F(c',c') \leftarrow W \to F(c,c) in DD, if the square on the left commutes, then the outer hexagon also commutes:

F(c,c) α c G(c,c) F(c,f) G(c,f) W F(c,c) G(c,c) F(f,c) G(f,c) F(c,c) α c G(c,c). \begin{array}{ccccccc} & & F(c,c) & \overset{\alpha_{c}}{\to} & G(c,c)\\ & \nearrow & & \overset{\mathclap{F(c,f)}}{\searrow} & & \overset{\mathclap{G(c,f)}}{\searrow}\\ W & & & & F(c,c') & & G(c,c')\\ & \searrow & & \underset{\mathclap{F(f,c')}}{\nearrow} & & \underset{\mathclap{G(f,c')}}{\nearrow}\\ & & F(c',c') & \underset{\alpha_{c'}}{\to} & G(c',c') \end{array}\,.

If the pullback F(c,c)× F(c,c)F(c,c)F(c,c) \times_{F(c,c')} F(c',c') exists in DD, it suffices to assert this when the square on the left is the defining one of that pullback. On the other hand, if DD has a separator (such as 1Set1\in Set), it suffices to assert this when WW is the separator.

By comparison, a dinatural transformation asserts this condition only when W=F(c,c)W = F(c',c) with the span consisting of F(c,f)F(c',f) and F(f,c)F(f,c).


Originally introduced (as strong dinatural transformations) in Definition 2.7 of

  • Philip S. Mulry?, Strong Monads, Algebras and Fixed Points (1991), in: Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, 202–216, doi.

Further developments:

  • Philip S Mulry. Strong monads, algebras and fixed points. Applications of Categories in Computer Science 177 (1992), 202–216

  • Robert Paré and Leopoldo Román. Dinatural numbers. Journal of Pure and Applied Algebra 128, 1 (1998), 33–92

  • A. Eppendahl, Parametricity and Mulry’s Strong Dinaturality,

  • Varmo Vene. Parametricity and Strong Dinaturality. (2006).

  • Jennifer Hackett and Graham Hutton. Programs for cheap!. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 115–126.

  • Tarmo Uustalu. Strong dinaturality and initial algebras. 12th Nordic Wksh. on Programming Theory, NWPT 2 (2000).

  • Tarmo Uustalu. A Note on Strong Dinaturality, Initial Algebras and Uniform Parameterized Fixpoint Operators. In FICS. 77–82.

Strong dinatural transformations are called paranatural transformations in:

Last revised on October 27, 2023 at 22:29:46. See the history of this page for a list of all contributions to it.