# Subcategory of Isomorphisms

There is a functor $\mathit{Iso} : \mathbf{Cat} \rightarrow \mathbf{Cat}$ which identifies the subcategory of a category in which only the isomorphisms appear as arrows — i.e. it strips off any arrow that does not have an inverse. This is because identities are isomorphisms, a composition of isomorphisms is an isomorphism, and functors preserve isomorphism.

My question is this: can this functor be constructed out of more familiar categorical concepts? For example, is it a limit of some diagram in $\mathbf{Cat}$? Also, is there a standard name for this functor?

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Presumably, this is just curiosity, so you have not seen it in nature yourself right? – Karl Kronenfeld Mar 28 '13 at 2:11
@user67848, In nature? I have seen it often (wearing another mask) when I am writing Haskell code. – luqui Mar 28 '13 at 3:32

This functor really takes values in the category of (small) groupoids $\text{Gpd}$. Once you've made that modification, it's the right adjoint to the inclusion functor $\text{Gpd} \to \text{Cat}$, which exhibits $\text{Gpd}$ as a coreflective subcategory of $\text{Cat}$. It is sometimes called the core.