nLab
closed category

Idea

A category $C$ is closed if for any pair $a,b$ of object the collection of morphisms from $a$ to $b$ can be regarded as forming itself an object of $C$.

This object is often denoted $\mathrm{hom}(a,b)$ or $[a,b]$ or similar and often addressed at the internal hom-object or simply the internal hom.

A familiar kind of closed categories are closed monoidal categories. However, there is also a definition of closed category that does not require the category to already be monoidal. A monoidal structure $\otimes$, if it exists, can then be universally characterized as a left adjoint to the internal-hom, dual to the above characterization of internal-homs as right adjoints to $\otimes$. See Eilenberg-Kelly, referenced below.

While this is less fashionable, in some cases it is more obvious what the correct internal-homs are than what the correct tensor product is, so the latter was originally defined as an adjoint to the former. This is the case for the Gray tensor product and was probably the case for abelian groups as well.

References

Closed categories were first defined here:

• Samuel Eilenberg and Max Kelly, Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).

You can get some of the idea from a post by Owel Biesel at the $n$-Café.