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For any double category <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {A}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>A<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the corresponding internal hom functor \"Equation missing\" sends a double category <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {B}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>B<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula> to the double category whose 0-cells are the double functors <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {A}} \\rightarrow {\\mathbb {B}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.\n<\/jats:p>","DOI":"10.1007\/s10485-019-09587-5","type":"journal-article","created":{"date-parts":[[2019,12,18]],"date-time":"2019-12-18T18:02:43Z","timestamp":1576692163000},"page":"477-515","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["The Gray Monoidal Product of Double Categories"],"prefix":"10.1007","volume":"28","author":[{"given":"Gabriella","family":"B\u00f6hm","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,12,18]]},"reference":[{"key":"9587_CR1","doi-asserted-by":"publisher","DOI":"10.1090\/crmm\/029","volume-title":"Monoidal Functors, Species and Hopf Algebras. 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