If your edited question is this:

Let $C$ be a category and let $W$ be the class of isomorphisms of $C$; note that $W$ is closed under composition and so defines a subcategory of $C$, which is the core of $C$. Must $(C,W)$ admit a calculus of right fractions?

then the answer is yes. (And the situation is symmetric, so it also admits a calculus of left fractions.)

]]>Certainly it is.

]]>Is the core of a category always a wide subcategory? I mean, in the most laziest sense of isomorphisms, why not include identity morphisms?

Edit: I’m curious if cores always have a notion of fraction.

]]>