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"?