A semi-strictly generated closed structure on Gray-Cat

We show that the semi-strictly generated internal homs of Gray-categories ${[A,B]}_{\text{ssg}}$ defined in [19] underlie a closed structure on the category Gray-Cat of Gray-categories and Gray-functors. The morphisms of ${[A,B]}_{\text{ssg}}$ are composites of those trinatural transformations which satisfy the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible 3-cell. Such trinatural transformations leverage three-dimensional strictification [19] while overcoming the challenges posed by failure of middle four interchange to hold in Gray-categories [3]. As a result we obtain a closed structure that is only partially monoidal with respect to [8]. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories [13], which will be improved further in a forthcoming paper [21].