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.



Localizing 2-categories about a single morphism
Well, if you are happy with everything maximally strict and finite, then you are really looking for a description of the coinverter (a weighted colimit) in the 1-category of finite 2-categories enriched over the cartesian monoidal category of finite 2-categories. See section 2.1 of [John Bourke' thesis](http://www.math.muni.cz/~bourkej/papers/JohnBThesis.pdf), in particular pages 13-14 (and work backwards for definitions). 


In your setup you want the coinverter of the 2-arrow
$$
f\colon x \Rightarrow y \colon \ast \to \mathbf{C}
$$
where $x$ and $y$ are the constant functors at the named objects, and the natural transformation has as only component the 1-arrow $f$. As Bourke mentions, this can be constructed from coinserters and coequifiers (these are described on previous pages). Since you are dealing with a particularly small class of arrows this may be not as tricky as it first sounds.


Note that this is not original to Bourke, and goes back to [Max Kelly](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html).


