Consider the following pullback diagram (in any category):

$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
A \times\_C B & \ra{p} & A \\
\da{q} & & \da{a} \\
B & \ras{b} & C \\
\end{array}
$$

with $a$ a monomorphism and $b$ an epimorphism. I would like to understand necessary and/or sufficient conditions for $p$ to be an epimorphism too.

The following might be relevant:

> 
> **Lemma:** $q$ is always a monomorphism (and dually, in all pushout diagrams, the same statement holds for the epimorphism)
> 
> 
> **Proof:** Assume that we have two maps $u\_1, u\_2$ from some object $W$ to $A \times\_C B$ such that $q \circ u\_1 = q \circ u\_2$. Then $a \circ p \circ u\_1 = a \circ p \circ u\_2$ because the diagram is Cartesian, and therefore $p \circ u\_1 = p \circ u\_2$ because $a$ in a monomorphism. But since $u\_1$ and $u\_2$ are uniquely determined by their compositions $p \circ u\_i$ and $q \circ u\_i$, and since both coincide, we have that $u\_1 = u\_2$, and therefore $q$ is a monomorphism. QED
> 
> 
> 

I have read [here](https://stacks.math.columbia.edu/tag/08N3) a proof for Abelian categories, but I think that my proof should be ok for any category, am I wrong? Anyway, back to the main point:

> 
> **Question:** When is $p$ an epimorphism too?
> 
> 
> 

I don't expect this to be always true. But maybe there are necessary and/or sufficient conditions for when this works. For instance, for Sets it is always true (just look at the explicit construction of the pullback).

PS: I have seen [here](https://stacks.math.columbia.edu/tag/08N4) that it is true in any Abelian category.