This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and assume the existence of a $1$-morphism $f:x \to y$ in $\mathbf{C}$ from some object $x$ to some object $y$ and moreover that there are no morphisms from $y$ back to $x$.

That is to say, $f$ is an object in the category $\mathbf{C}(x,y)$ of morphisms from $x$ to $y$ while the category $\mathbf{C}(y,x)$ is empty. To avoid set-theoretic issues, one can safely assume that $\mathbf{C}$ is hopelessly finite: there are finitely many objects, finitely many $1$-morphisms between any pair of objects, and finitely many $2$-morphisms between any pair of $1$-morphisms. Here's the question:

Is there a clean description of the $2$-cells in the localization $f^{-1}\mathbf{C}$ of $\mathbf{C}$ at the single morphism $f$?

I know that the objects are the same as those of $\mathbf{C}$, and it is clear that the $1$-morphisms in $f^{-1}\mathbf{C}$ from $w$ to $z$ are given by formally augmenting the old morphisms $\mathbf{C}(w,z)$ with "new ones" of the form $gf^{-1}h$ where $g:w \to y$ and $h:x \to z$. But what are the $2$-cells? I can describe them in an unweildy case-by-case manner (i.e. old-to-new, new-to-new,...) but was hoping for something nicer that can also be used as a reference.

I'd be happy with a reference that handles the more general case of localizing about a larger collection $W$ of morphisms, but what I am *not* looking for is the case when $W$ forms a multiplicatively closed set.