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While editing rng I discovered that there was already a somewhat orphaned page unitalization. I have now made plenty of variants of this term redirects to that page (“unitisation” etc.) and have cross-linked with existing entries such as unitisation of C*-algebras.
added to unitalization a pointer to prop. 5.2.3.13 in Higher Algebra, which gives unitalization in the generality of $\mathbb{E}_k$-algebras.
It's especially difficult to get things linked properly where there are unexpected spelling variations, so I'm glad that you caught that!
The unitalization functor is not a conservative functor. \cite{Andruszkiewicz}
However, it does become conservative when restricted to the full subcategory of nonunital algebras over a field $k$ that do not admit a surjective homomorphism to $k$. \cite{Andruszkiewicz}
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