This question is about (not necessarily symmetric) monoidal categories enriched over a symmetric monoidal category $\mathcal{V}$. Assume that $\mathcal{V}$ is closed. You may also assume that $\mathcal{V}$ is (co)complete if you wish.
If $k$ is a commutative ring, a $k$ algebra can be defined in two ways. Either as a $k$-module $R$ together with morphisms $k\rightarrow R$ and $R\otimes_{k}R\rightarrow R$ satisfying the well-known laws, or as a ring homomorphism to the center $k\rightarrow Z(R)$.
Let's see what happens in the categorical context.
The tensor product of $\mathcal{V}$-enriched categories can be straightforwardly defined, see Kelly's book. Then one can define what a monoidal $\mathcal{V}$-category is by reproducing the classical definition in the enriched context.
Assume now that $\mathcal{C}$ is an ordinary monoidal category. I believe that the braided center $Z(\mathcal{C})$ as defined by Joyal and Street is a well known construction. Suppose that we have a strong braided monoidal functor $z : \mathcal{V}\rightarrow Z(\mathcal{C})$ such that the functor $z(-)\otimes Y : \mathcal{V}\rightarrow \mathcal{C}$ has a right adjoint ${Hom}_{\mathcal{C}}(Y,-) : \mathcal{C}\rightarrow\mathcal{V}$ for any object $Y$ in $\mathcal{C}$. The counit is an evaluation morphism in $\mathcal{C}$,
$ev: z( {Hom}_{\mathcal{C}}(Y,Z))\otimes Y\longrightarrow Z$
One can define composition morphisms in $\mathcal{V}$
${Hom}(Y,Z)\otimes {Hom}(X,Y)\longrightarrow {Hom}_{\mathcal{C}}(X,Z) $
as the adjoint of
$z({Hom}(Y,Z)\otimes {Hom}(X,Y))\otimes X \cong
z({Hom}(Y,Z))\otimes z({Hom}(X,Y))\otimes X
\stackrel{id \otimes ev}\longrightarrow
z({Hom}(Y,Z))\otimes Y
\stackrel{ev}\longrightarrow
Z $
I think it's pretty obvious that $\mathcal{C}$ becomes $\mathcal{V}$-enriched in this way. Moreover, one can also enrich the tensor product in $\mathcal{C}$ in a similar way.
Do you guys agree? Do you know of any reference where this is checked with some detail? Is it even more obvious than I think?
Any comment is welcome!
Best Answer
There is a theorem in category theory, generally regarded as folklore, which says that for a symmetric monoidal closed category $V$, the following structures are equivalent:
You can see this, for example, in the appendix to this paper.
In your case, unless I've misunderstood, the centre $Z(C)$ plays little role. The point is that your functor $z:V\to C$ induces an action via $v*c=z(v)\otimes c$, and $-*c$ has a right adjoint by assumption, so you get the $V$-enrichment.
(There is an analogous characterization of $V$-categories $C$ which are cotensored/powered: this means that each $C(-,d):C^{op}\to V$ has a left adjoint.)