Descent in Locally Presentable Categories

Let 𝕍=(VV,⊗,I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {V}=(VV, \otimes , I)$\end{document} be a symmetric monoidal category such that 𝒱\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {V}$\end{document} is locally presentable and that all functors V⊗−:𝒱→𝒱\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V\otimes - : \mathcal {V} \rightarrow \mathcal {V}$\end{document} for V∈𝒱\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V \in \mathcal {V}$\end{document} preserve reflexive coequalizers and directed colimits. It is proved that any pure morphism of commutative 𝕍-monoids is an effective descent morphism with respect to the indexed category given by commutative 𝕍-monoids and modules over them. As a by-product, we prove that pure morphisms in a locally presentable category are effective for codescent.