**SymMonCat** is the cartesian 2-category of symmetric monoidal categories, braided monoidal functors, and monoidal natural transformations. The terminal symmetric monoidal category **1** has one object $I$ and $I \otimes I = I$.


A category enriched over a monoidal category $V$ assigns to each pair of objects $X, Y$ an object hom$(X,Y)$ in $V$ and to each object $X$ a morphism $id\_X:I \to \mbox{hom}(X,X)$ in $V$.


When $V = $ **SymMonCat**, the morphism $id\_X:1 \to \mbox{hom}(X,X)$ is a braided monoidal functor; since monoidal functors preserve the monoidal unit and tensor product, it must map the unit $I$ in **1** to the unit $I$ in hom$(X,X)$.


Is there a different way of enriching over **SymMonCat** such that $id\_X$ does not pick out the monoidal unit (other than considering it a subcategory of **Cat**)? 



Is there a sensible way to enrich over SymMonCat such that id_X is not the monoidal unit?
[Ignore this first part, I'm just leaving it for the context to the comments below.] 
It is hard for me to understand why you would want to enrich in symmetric monoidal categories, have an identity, and also want this identity to *not* be the unit of the symmetric monoidal category. 


That said, you can always do away with units altogether and consider "enriched categories without identities". Is this what you are after?




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After Mike's example I am now on board. What you probably want to do is enrich over the symmetric monoidal 2-category of symmetric monoidal categories where the monoidal structure is the "tensor product of symmetric monoidal categories". What is this you ask? 


The functor category between two symmetric monoidal categories $Fun^\otimes(B,C)$ is naturally equipped with a symmetric monoidal structure (using pointwise multiplication). The tensor product of symmetric monoidal categories is $(-) \otimes B$ is the (weak) left adjoint to the functor $Fun(B, -)$. Thus $A \otimes B$ is a symmetric monoidal category such that symmetirc monoidal functors from it to C are the same as "bilinear" functors $A \times B \to C$. Now the monoidal unit for this tensor product is the free symmetric monoidal category on one object $\mathbb{F}$ (which is the category of finite sets and permutations). 


In this way, if you enrich in (SymCat, $\otimes$) you get a unit being a functor $ \mathbb{F} \to Hom(a,a)$, which is equivalent to just some element, not necessarily the unit object of $Hom(a,a)$. 


The prototypical example is the 2-category of symmetric monoidal categories itself. 


