Let $\mathbb{C}$ be a self-enriched category (such as Set).

The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$.

In a similar manner, how can the functor $\mathbb{C}(\mathbb{C}(X,\_), \mathbb{C}(\mathbb{C}(Y,\_),\_))$ be rewritten as a composition of hom-functors?

  • $\begingroup$ Do you want $\mathbb C(\mathbb C(X, \_), \mathbb C(\mathbb C(Y, \_), \_)) (D) = \mathbb C(\mathbb C(X, D), \mathbb C(\mathbb C(Y, D), D))$? $\endgroup$ – fauxefox Apr 3 at 20:45
  • $\begingroup$ @fauxefox Yes, that's what I mean. $\endgroup$ – Bob Apr 3 at 20:55
  • $\begingroup$ Unless $\mathbb C$ is self-enriched, such as $\mathbf{Set}$, these expressions don't make sense. $\endgroup$ – Derek Elkins Apr 3 at 21:53
  • $\begingroup$ @DerekElkins Indeed. The case of interest to me in Set but I would prefer an abstract solution for any self-enriched category. I have edited my question accordingly. Do you have an answer? $\endgroup$ – Bob Apr 3 at 22:00

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