TITLE: When the set of morphism between two objects is an object in the category?
QUESTION [5 upvotes]: In the category of R-Modules, for a given ring R, The set of all morphism between two objects is an object in the category. Obviously this is also true in the category of sets.
Are there more categories in which this is true?

REPLY [3 votes]: In addition to the information provided by Mike Haskel about closed monoidal categories (and the more general notion of monoidal categories), you should also look into the more general notion of closed categories. These have an internal Hom functor $[-,-]$ but need not have a left adjoint $-\otimes A$ for $[A,-]$. So they match more closely the exact formulation in the question.  The theories of both closed and monoidal categories were presented in considerable detail by Eilenberg and Kelly in a paper "Closed categories" from 1966; Fred Linton's review on MathSciNet at MR0225841 (37 #1432) gives a good overview of the central ideas.