Lemma 5.30.12. Let $R$ be a topological ring. The category of topological modules over $R$ has colimits and colimits commute with the forgetful functor to the category of modules over $R$.
Proof. The exact same argument as used in the proof of Lemma 5.30.6 shows existence of colimits. To see the statement on commutation with the forgetful functor to $R$-modules we will use Categories, Lemma 4.24.5. Indeed, the forgetful functor has a right adjoint, namely the functor which assigns to a module the corresponding chaotic (or indiscrete) topological module. $\square$
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