Q:

Pullback preserves cokernel

Is that true that in an abelian category $\mathcal{C}$, if I have the pullback diagram:
$$
\require{AMScd}
\begin{CD}
P @>{p_1}>> C\\
@V{p_2}VV @V{g}VV \\
A @>{f}>> B
\end{CD}
$$
with $f$ and $p_1$ monomorphisms, then $Coker(f) = Coker(p_1)$? 

A:

Nope!  Let $C = 0$, for example. 
What is true is that $\text{Coker}(p_1) \to \text{Coker}(f)$ is a monomorphism. 
(it doesn't matter whether or not $f$ and $p_1$ are monomorphisms)

