Given functions $f : X \rightarrow Y$ and $g : Y \rightarrow X,$ we say that $f$ and $g$ are *inverses* iff the following holds.

$$fx = y \Leftrightarrow x=gy$$

We can rehash this condition in terms of pullbacks. Given functions $p,q: A,B \rightarrow C,$ define that their pullback is $$p \times\_C q = \{(a,b) \in A \times B \mid pa = qb\},$$

in which case it follows that any two functions $f : X \rightarrow Y$ and $g : Y \rightarrow X$ are inverses iff $$(\*)\quad f \times\_Y \mathrm{id}\_Y = \mathrm{id}\_X \times\_X g.$$

Now as Berci points out, the above pullbacks exist in any category.

**Question:** Is there a way to phrase $(\*)$ so that it makes sense in arbitrary categories?