This construction extends to a 1-functor
Core \colon Cat \to Grpd \,.
We usually think of a groupoid as a special kind of category, but we can also think of a category as a groupoid equipped with additional morphisms. (This is possible because Grpd is a reflective subcategory of Cat.) One level decategorified, we usually think in the opposite way: a poset is a set equipped with a partial order, but we can also think of a set as a special kind of poset (specifically, a symmetric one).
A \to C
(hence a functor out of the underlying category of ) has to send isomorphisms to isomorphisms, hence has to send every morphism of to an isomorphism in . This means that it factors through the core-inclusion
A \to Core(C) \to C \,.
The core of a dagger category consists of its unitary isomorphisms only. This is why, for example, it makes sense to think of Hilb either as a category whose morphisms are linear maps bounded by or as a dagger category whose morphisms are all linear maps; either way, the core is the same (invertible linear maps of norm exactly ).
The core of an -category is the -groupoid consisting only of equivalences at each level; the core of an -category is similarly an -groupoid: the core of a quasicategory is the maximal Kan complex inside it.
For more on this see also at category object in an (infinity,1)-category.