Pullback preserves cokernel

Question

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 $Coker(f) = Coker(p_1)$?

Answer 1

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)