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!

`sciencedirect.com`

is broken, but the article can be found at doi:10.1016/0022-4049(91)90039-5 (Zbl 0726.18004). $\endgroup$