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I know this question may seem nonsensical at first but let me exlain what i have in mind:

In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, I)$. An enriched category then is given by a set/class of objects $\mathcal C$ and a rule assigning to every pair $X,Y$ of such objects a hom-object $[X,Y]$. Furthermore we define composition and identities using $\otimes$ and $I$, remodelling the definitions of usual category theory.

Now for the question: Can we go the other way around?

Let's stick to internal homs for the beginning: Given a category $\mathcal V$ ; can say what additional data turn a functor $$[-,-]:\mathcal{V}^{\mathrm{op}}\times\mathcal V\to \mathcal V$$ into something like an internal hom?

In the case of $[X,-]$ having a left adjoint $-\otimes X$ for every $X$, these additional data should result in $(\mathcal V,\otimes)$ becomming a closed monoidal category with internal hom isomorphic to $[-,-]$.

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This is exactly the notion of a closed category. See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed category.

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    $\begingroup$ If I recall correctly, La Plaza showed that every closed category in this sense can be fully and faithfully embedded in a monoidal closed category in a way respecting the closed structure. $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2011 at 19:08
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    $\begingroup$ The Laplaza paper is in Trans. AMS 233, 1977. $\endgroup$
    – Finn Lawler
    Commented Oct 25, 2011 at 19:36
  • $\begingroup$ Note that closed categories are not quite strong enough to recover monoidal categories: we must additionally impose an associativity constraint, as described in this answer. $\endgroup$
    – varkor
    Commented Jun 8, 2023 at 20:11

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