The internal hom (or exponential object) is basically a reification of the 'external' hom. It can be defined in any cartesian (or even monoidal, more on this later) category as the right adjoint of the (monoidal) product.

My question is: are there some categories whose internal hom behaves quite unexpectedly? Or even some interesting examples where internalization really deforms the external hom in a non-trivial way.

The question arises from the observaton that the 'external' hom can be assumed to be a set (assume the category to be locally small, or wave your hand hard enough), and thus is already somewhat reified. We expect some 'structure' on it, namely elements and perhaps even subobjects. In a sufficiently rich category (say a topos), this structure is internalizable as well. So it might be the case that some exotic behavior emerge, or some collapse happens.

I expect this to happen mainly for monoidal closed category, because the monoidal product can be quite involved.

On the other hand, Section 3 of the internal hom page of the nLab seems to prove the internal hom shares some strong properties of the 'external' hom, which might hint to the fact they are really 'the same'.

The internal hom (or exponential object) is basically a reification of the 'external' hom. It can be defined in any cartesian (or even monoidal, more on this later) category as the right adjoint of the (monoidal) product.

My question is: are there some categories whose internal hom behaves quite unexpectedly? Or even some interesting examples where internalization really deforms the external hom in a non-trivial way.

The question arises from the observaton that the 'external' hom can be assumed to be a set (assume the category to be locally small, or wave your hand hard enough), and thus is already somewhat reified. We expect some 'structure' on it, namely elements and perhaps even subobjects. In a sufficiently rich category (say a topos), this structure is internalizable as well. So it might be the case that some exotic behavior emerge, or some collapse happens.

I expect this to happen mainly for non-Cartesian monoidal closed category, because the monoidal product can be quite convoluted.

On the other hand, Section 3 of the internal hom page of the nLab seems to prove the internal hom shares some strong properties of the 'external' hom, which might hint to the fact they are really 'the same'.