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Sep 2, 2019 at 21:16 comment added Tim Campion Or representations of a group $G$ (or Hopf algebra more generally). Similar to chain complexes or graded abelian groups, the internal hom consists of all linear maps -- not just $G$-equivariant ones.
Aug 19, 2019 at 16:00 comment added David White Another answer, in the spirit of chain complexes, is the internal hom of Spectra (symmetric, orthogonal, or S-modules), G-spectra, and motivic spectra
Aug 18, 2019 at 14:44 comment added Peter LeFanu Lumsdaine @SimonHenry: why not make that an additional answer, since it has quite a distinct flavour from the examples here?
Aug 18, 2019 at 14:24 comment added Simon Henry The category of set and relation (monoidal for the products of sets) is also a good example. The exponential is given by the product of sets.
Aug 18, 2019 at 14:03 history edited Peter LeFanu Lumsdaine CC BY-SA 4.0
fixed typo
Aug 18, 2019 at 12:02 history answered Peter LeFanu Lumsdaine CC BY-SA 4.0