How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is? In some algebraic categories (abelian groups, modules on a ring) and in pointed spaces categories the following holds:
$$ [X,[Y,Z]] \cong [X \times Y, Z] $$
In wich generality this lemma holds?
 A: The general notion is of a closed monoidal category, which is essentially a category equipped with a notion of tensor product $- \otimes -$ and an "interal hom" $[-, -]$ between any two objects that behaves like an "object of morphisms", such that tensor product is left adjoint to internal hom, i.e., $\operatorname{Hom}(A \otimes B, C) = \operatorname{Hom}(A, [B, C])$ for all objects $A, B, C$.
The linked article lists many examples, but to add one to the list, consider the tensor product and "sheafy hom" of sheaves of modules on a ringed space $(X, \mathcal{O}_X)$: Given sheaves of $\mathcal{O}_X$-modules $F$ and $G$, we define $\mathcal{H}om(F, G)$ to be the functor $U \mapsto \operatorname{Hom}_{\mathcal{O}_X \rvert_U}(F \rvert_U, G \rvert_U)$, which defines a sheaf of $\mathcal{O}_X$-modules; and define $F \otimes_{\mathcal{O}_X} G$ to be the sheafification of the presheaf $U \mapsto F(U) \otimes_{\mathcal{O}_X(U)} G(U)$. These satisfy a tensor-hom adjunction $\operatorname{Hom}_{\mathcal{O}_X}(F \otimes_{\mathcal{O}_X} G, H) = \operatorname{Hom}_{\mathcal{O}_X}(F, \mathcal{H}om(G, H))$. When $X = \{*\}$ is the one-point space and $\mathcal{O}_X(*) = R$, this reduces to the example of the category of $R$-modules.
