# The Gray monoidal product of double categories

## TL;DR

This paper establishes a symmetric closed monoidal structure on the category of double categories, defining an internal hom functor that captures double functors, pseudotransformations, and modifications, with practical functors shown to be compatible.

## Contribution

It introduces a new symmetric closed monoidal structure on double categories and characterizes the internal hom functor with compatibility results for key functors.

## Key findings

- Defines the symmetric closed monoidal structure for double categories.
- Constructs the internal hom functor with detailed cell structures.
- Shows compatibility of well-known functors with this monoidal structure.

## Abstract

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double category whose 0-cells are the double functors $\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.