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"?

Basic concepts of enriched category theorytac.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 18:29usualway since it is cartesian closed. But it also hasanotherself-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