Suppose I define a multicategory $M=(Ob(M),Hom\_M)$ to be *simply closed* if 


* for every sequence $S=(b\_1,\ldots,b\_n;x)$ of $n+1$ objects in $M$, we provide an object $Exp(S)\in Ob(M)$, and
* for every sequence $A=(a\_1,\ldots,a\_m)$ of $m$ objects in $M$ we provide a bijection
$$\phi\_S^A\colon Hom\_M(a\_1,\ldots,a\_m,b\_1,\ldots,b\_n;x)\to Hom\_M(a\_1,\ldots,a\_m;Exp(b\_1,\ldots,b\_n;x)),$$


subject to the condition below, for any object $x\in Ob(M)$. Fix such an object $x\in Ob(M)$ and let $X:=Exp(;x)\in Ob(M)$. (Apologies: fixing $x$ and introducing $X$ is done here only for typographical reasons: mathjax was having trouble rendering.) We name a "constant element" $c\_x\in Hom\_M(x;X)$ by setting $A=(x)$ and $S=(;x)$, so $\phi^A\_S\colon Hom\_M(x;x)\to Hom\_M(x;X)$, and putting 
$$c\_x:=\phi^{(x)}\_{(;x)}(id\_x)\in Hom\_M(x;X).$$ 
We also name an "evaluation element" by setting $A=(X)$ and $S=(;x)$, so $(\phi^A\_S)^{-1}\colon Hom(X;X)\to Hom(X;x)$, and putting 
$$e\_x:=\left(\phi^{(X)}\_{(;x)}\right)^{-1} (id\_X)\in Hom\_M(X;x).$$
Then our condition is that the constant element and the evaluation element for $x$ are mutually inverse in the sense that 
$$c\_x\circ e\_x=id\_X \;\;\;\;\;\;\text{ and }\;\;\;\;\;\; e\_x\circ c\_x= id\_x$$ in $M$.


As alluded to above, we obtain an ``evaluation map" as an element $ev\in Hom\_M(Exp(A;x),A;x)$, coming as $\phi^{-1}(id)$, where $id$ is the identity element $id\in Hom\_M(Exp(A;x),Exp(A;x))$. 


The multicategory of sets is simply closed in this sense.


What problems exist for, and/or what niceties are missing from, this notion of "simply closed multicategory"?



A (too?) simple notion of "closed multicategory"
In fact, if you include naturality, then your second condition becomes automatic, and moreover it suffices to consider $n=1$. See e.g. section 3 of [this paper](http://arxiv.org/abs/0904.3137).


