# decomposition of hom-functors in a self-enriched category

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?

• Do you want $\mathbb C(\mathbb C(X, \_), \mathbb C(\mathbb C(Y, \_), \_)) (D) = \mathbb C(\mathbb C(X, D), \mathbb C(\mathbb C(Y, D), D))$? – fauxefox Apr 3 at 20:45
• @fauxefox Yes, that's what I mean. – Bob Apr 3 at 20:55
• Unless $\mathbb C$ is self-enriched, such as $\mathbf{Set}$, these expressions don't make sense. – Derek Elkins Apr 3 at 21:53
• @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? – Bob Apr 3 at 22:00