This is probably an easy question. Let C be a category with (finite) products.
 An internal hom in C category is an object uhom(X, Z) which represents the functor:


Y |-----> hom(Y x X, Z)


here "uhom" is for "underlined hom" as that is how it is commonly denoted. Many example of categories with internal homs satisfy an a priori stronger for of adjunction:


uhom(Y x X. Z) = uhom(Y, uhom(X,Z))


Is this automatic for categories with internal homs? Is there an easily understood counter example?


(\*) This might not be the most general/best definition of internal hom, but it is valid for many examples. 



internal homs and adjunctions?
I think both uhom(Y × X, Z) and uhom(Y, uhom(X, Z))
represent the same functor:
hom(W, uhom(Y × X, Z)) = hom(W × Y × X, Z)
hom(W, uhom(Y, uhom(X, Z))) = hom(W × Y, uhom(X, Z)) = hom(W × Y × X, Z),
hence by Yoneda lemma they are isomorphic.


As a side remark, the right notion of inner hom in the absence of monoidal structure is the notion of closed category:
<http://en.wikipedia.org/wiki/Closed_category>


