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Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor $$ X \otimes -: \cal{C} \to \cal{C}, $$ for each object $X$ in $\cal{C}$.

Denoting this functor by $$ hom_X(-):\cal{C} \to \cal{C} $$ it is tempting to ask if the functor $$ hom: {\cal C} \times {\cal C} \to {\cal C}, ~~~~~~~ (X,Y) \mapsto hom_X(Y), $$ gives an enrichment of $\cal{C}$ over itself. Is this correct? Moreover, is the existence of an enrichment of $\cal{C}$ over itself equivalent to the existence of internal homs?

More generally, when people speak of internal homs for a category, not necessary monoidal, are they just talking about an enrichment of the category over itself? Is this what usually understood by "a category with internal homs"?

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    $\begingroup$ Yes, a closed monoidal category is enriched over itself. In the symmetric case this is section 1.6 of the standard reference Basic concepts of enriched category theory tac.mta.ca/tac/reprints/articles/10/tr10abs.html. I think that usually when people say "internal homs" they are referring to a monoidal structure being closed, or more generally to a closed category structure (ncatlab.org/nlab/show/closed+category) -- it doesn't make sense to talk about enrichment over a category until it has a monoidal/closed/multi/etc structure. $\endgroup$ – Mike Shulman Jun 20 '18 at 18:29
  • $\begingroup$ I would expect that a monoidal category $C$ could happen to be the underlying category of some $C$-enriched category in a "wrong" way so that the enriched hom-objects aren't right adjoints of the tensor product, but I don't have an example ready to hand. $\endgroup$ – Mike Shulman Jun 20 '18 at 18:30
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    $\begingroup$ Here's an example of "wrong-way" self-enrichment: the category $Cat$ of small categories is enriched in itself in the usual way since it is cartesian closed. But it also has another self-enrichment where you take the maximal subgroupoid of each hom-category. This sort of variant enrichment is important e.g. to make $Cat$ into a simplicial model category. $\endgroup$ – Tim Campion Jun 21 '18 at 13:21
  • $\begingroup$ @Tim: Put this as an answer and I can accept it. $\endgroup$ – Max Schattman Jun 21 '18 at 14:50
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As Mike says, it is indeed the case that a closed monoidal structure gives rise to a self-enrichment of a category. This is an often-used fact.

Here's an example of "wrong-way" self-enrichment: the category $\mathsf{Cat}$ of small categories is enriched in itself in the usual way since it is cartesian closed. But it also has another self-enrichment where you take the maximal subgroupoid of each hom-category. This sort of variant enrichment is important e.g. to make $\mathsf{Cat}$ into a simplicial model category.

For that matter, $\mathsf{Cat}$ also has a self-enrichment where you take the usual $\mathsf{Set}$-enrichment, and regard the hom-sets as discrete categories.

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I see that this question has an already accepted answer but I think that may be of interest.

There is a notion of category with internal hom with no reference to a monoidal structure, that is the notion of a closed category.

This is a category with an internal-hom functor and few other natural and extranatural transformations satisfying coherence conditions.

The best reference on the subject is Eilenberg and Kelly's Closed categories. In this paper the authors show how you can enrich in any closed category, without any monoidal structure, and personally I think is one of the best references for enriched categories (even if it is not the most complete).

I would strongly suggest to take a look to the article, personally I have learned and understood more enriched-category theory from that paper than other reference.

I hope this may help.

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