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We give a direct proof that, provided that the codensity monad of a morphism <jats:italic>p<\/jats:italic> exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the <jats:italic>two-dimensional cokernel diagram<\/jats:italic> of <jats:italic>p<\/jats:italic> is up to isomorphism the same as the semantic factorization of <jats:italic>p<\/jats:italic>, either one existing if the other does. The result can be seen as a counterpart account to the celebrated B\u00e9nabou\u2013Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of <jats:italic>p<\/jats:italic> trivially hold whenever <jats:italic>p<\/jats:italic> has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on <jats:italic>p<\/jats:italic>, namely, to be an effective faithful morphism of the 2-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\">\n                  <mml:mi>A<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10485-022-09694-w","type":"journal-article","created":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T09:02:41Z","timestamp":1668502961000},"page":"1393-1433","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Semantic Factorization and Descent"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1817-2797","authenticated-orcid":false,"given":"Fernando","family":"Lucatelli Nunes","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"issue":"5","key":"9694_CR1","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1007\/s10485-018-9530-6","volume":"26","author":"J Ad\u00e1mek","year":"2018","unstructured":"Ad\u00e1mek, J., Sousa, L.: A formula for codensity monads and density comonads. 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