Examples of categories where epimorphism does not have a right inverse, not surjective

An epimorphism is defined as follows:

$f \in \operatorname{Hom}_C(A,B)$ is an epimorphism if $\forall Z, \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ then $h' f = h'' f \; \Rightarrow \; h' = h''$.

I can't think of examples where epimorphism would not have a right inverse.

Also, if I understand correctly, epimorphism is not surjective in the categories where we can't talk about surjection (where objects does not have internal structure?).

Thanks in advance.

Take the category of a partially ordered set; every arrow is an epimorphism, but no non-identity arrow has a right inverse.

For a concrete category (objects are sets and morphisms are functions between the underlying sets), take the category of Hausdorff topological spaces; an epimorphism is a continuous map with dense image. Consider $\mathbb{Q}\hookrightarrow \mathbb{R}$. This is an epimorphism, but there is no retract (no right inverse). Or the map $[0,2\pi)\to S^1$ given by $t\mapsto (\cos t,\sin t)$. If it had a right inverse in the category, the inverse would be a bijection, hence we would have homeomorphisms, but $[0,2\pi)$ and $S^1$ are not homeomorphic.

For yet another, take $\mathbb{Z}\hookrightarrow\mathbb{Q}$ in the category of rings with unity. This is an epimorphism, but does not have a right inverse.

Even in concrete categories where all epis are surjective, you need not have right inverses. In the category of groups, an epimorphism $G\to K$ has a right inverse if and only if $G$ is a semidirect product $G\cong N\rtimes K$. So take $\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ for a surjective morphism (hence an epi) with no right inverse in the category.

Let $C$ be the category containing one object $\mathbb Z$, and morphisms being functions $f_n(z)=nz$ for $n\in\mathbb Z^+$. Then $f_n\circ f_m = f_{nm}$, and we can see that:

$$f_n\circ f_k = f_n\circ f_l \implies f_k=f_l$$

And similarly:

$$f_k\circ f_n = f_l\circ f_n \implies f_k=f_l$$

So every $f_n$ is an epimorphism and a monomorphism, but only $f_1$ has a left or right inverse. Note that $f_n$ is never surjective, if $n>1$, even though it is an epimorphism.

If you defined $C'$ in the same way, but with the object being $\mathbb Q$ and, for $n\in\mathbb Z^+$, $f_n(q)=nq$ is defined as a function on $\mathbb Q$, then this new category is (in some sense) isomorphic to $C$, and all functions are surjective and injective.