In general if you don't want to start with the monoidal structure, you start with a **closed category**. In a closed category $\mathsf C$, you have a bifunctor
$$[-,-] : \mathsf C^{op} \times \mathsf C \to \mathsf C,$$
called the internal hom, and various other data that are somewhat "dual" to the axioms of a monoidal category (I put dual in quotes because this isn't the dual notion of "monoidal category", you just dualize one side of the bifunctor $- \otimes -$ to get the axioms for $[-,-]$). More specifically:

- a unit object $I$,
- a natural isomorphism $\operatorname{id}_{\mathsf C} \cong [I, -]$,
- an extranatural transformation $j_X : I \to [X, X]$ that corresponds to getting the "identity" in $[X,X]$,
- post-composition transformation $[Y,Z] \to [[X,Y], [X,Z]$,

subject to various coherence conditions.^{*}

Then just like the internal hom $[-,X]$ in a monoidal category, if it exists, is the right adjoint to $- \otimes X$, the tensor product in a closed category is the left adjoint to the internal hom (if it exists). In both cases you get a closed monoidal category. It's also possible to define categories enriched over closed categories, in a manner similar to categories enriched over a monoidal category, and the two notions coincide when the category is closed monoidal.

^{*} These data are truly dual to the data you want for a monoidal category. Take the point of view that $\hom(X \otimes Y, Z) \cong \hom(X, [Y,Z])$ as if you were truly in a closed monoidal category. Then you can use the Yoneda embedding to make the correspondence clear:

- The isomorphism $[I, X] \cong X$ becomes $\hom(Y, [I,X]) \cong \hom(Y \otimes I, X)$ naturally in $X$ and $Y$, so you get the right unitor $- \otimes I \cong \operatorname{id}_{\mathsf C}$.
- The transformation $I \to [X,X]$ becomes a transformation $$\hom(Y, I) \to \hom(Y, [X,X]) \cong \hom(Y \otimes X, X).$$ If you take $Y = I$ you get $\hom(I,I) \to \hom(I \otimes X, X)$, and the image of the identity of $I$ because the left unitor.
- The data of the associator is equivalent to an (extra in some variables)natural isomorphism $[A, [B,C]] \cong [A \otimes B, C]$. The relation with the post-composition is trickier and I'll let you peruse the reference.

Of course none of the above is an actual proof, just a sketch of the ideas that go into this. Proofs can be found in:

Kelly, G. Max; Mac Lane, Saunders, *Coherence in closed categories*. J. Pure Appl. Algebra 1 1971 no. 1, 97–140. doi: 10.1016/0022-4049(71)90013-2.