TITLE: Examples of coregular categories
QUESTION [5 upvotes]: I'm interested in some examples of coregular categories (and a sketch of a proof, that they are in fact coregular). It would be nice to apply things I know about regular categories to some coregular categories too. 
Obviously, I'm not interested in things like: the dual of some topos, the dual of some algebraic category, an Abelian category; unless they go by some specific name.
Remark: This answer lists some examples, but without proof. Also it is unclear to me whether $\mathsf{Grp}$ is coregular or not, which was stated somewhat like an exercise there.

REPLY [4 votes]: $\mathsf{Grp}$  is not coregular: 
Let $K$ be a group with $N$ one of its nontrivial subgroup (that is, $N\unlhd K$ and $N$ is not $K$ or $1$), $G$ a simple group containing $K$. Let $f:K\to G$ be the canonical embedding and $g:K\to K/N$ be the canonical projection. 
For any $X$, $\varphi:G\to X$ and $\psi:K/N\to X$ such that $\varphi f=\psi g$, since  $\varphi(N)=\varphi f(N)=\psi g(N)=1$, $\varphi$ is not injective. But $G$ is simple, so we must have $\varphi$ is trivial, which forces that $\psi g$ ($=\varphi f$) is trivial. This implies that $\psi$ is trivial for $g$ is an epimorphism. In conclusion, the pushout of $f$ and $g$ is $1$.