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Let $\mathcal{C}$ be a category. I want to define another category $\mathcal{C}'$ putting ${\rm Obj}(\mathcal{C}') \doteq {\rm Obj}(\mathcal{C})$ and $${\rm Hom}_{\mathcal{C}'}(A,B) \doteq \{f \in {\rm Hom}_{\mathcal{C}}(A,B) \mid f \mbox{ is an isomorphism}\},$$for all $A,B \in {\rm Obj}(\mathcal{C}')$. Meaning I want to forget every morphism which is not an isomorphism.

Does $\mathcal{C}'$ have a standard name or a standard notation, in terms of $\mathcal{C}$?

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It is called the core of $\mathcal C$.

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