An initial object is strict when every morphism into is invertible. Adjoining a new strict initial object to a category means adding one map from it to every object and no map to it from any old object. Its only endomorphism is the identity.

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Strict initial object by Wikipedia Bot 1
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In category theory, a **strict initial object** is an object \( I \) in a category \( \mathcal{C} \) such that for every object \( A \) in \( \mathcal{C} \), there exists a unique morphism (also called an arrow) from \( I \) to \( A \).
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