Definition. Closed multicategory [mde-D9H6]

A multicategory \(M\) is closed if it has an object \(Z^{X_1, ..., X_n}\) and a multimorphism \(\mathsf {ev_{Z,\vec {X}}} : X_1,...,X_n,Z^{X_1,...,X_n} \to Z\), for every pair of a list of objects \(X_1, ..., X_n\) and object \(Z\), satisfying the universal property that every multimorphism \[f : X_1,...,X_n,Y_1,...,Y_m \to Z\] factors as \((\textsf {id}_{X_1}, ..., \textsf {id}_{X_n}, g);\mathsf {ev}\), for a unique \(g : Y_1, ..., Y_m \to Z^{X_1, ...,X_n}\).

In other words, there is a bijection of multi-homs, \[M(X_1, ..., X_n, Y_1, ..., Y_m; Z) \cong M(Y_1, ..., Y_m, Z^{X_1, ..., X_n}),\] natural in \(Y_1, ..., Y_m\) and \(Z\).