Enriched adjoint functor theorem?

[Gabor Lukacs <dr.gabor.lukacs@gmail.com>]

Dear All,

I was wondering if there is a known generalization of the adjoint functor
theorem to enriched categories. This is what I am trying to figure out:

I have a closed category V with internal hom-functpr [-,-], and I am
trying to show that it is *monoidal* closed. I was able to prove (using
the adjoint functor theorem) that the hom-functor [a,-] has a left-adjoint
L^a: V --> V, but in order to obtain a monoidal closed strucutre, one
needs to have a natural isomorphism in V:

[L^a(b),c] -=- [b,[a,c]]   (*)

This will also imply associativity and coherence.

So, I am asking if there is a way to prove (*) based on some form of
enriched adjoint functor theorem, without figuring out the structure
of L^a(b) explicitly.

Best,
Gabi

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