Is the fibre-product functor $(-)\times_N M$ exact?

Question

Consider an abelian category $\mathcal C$; if it helps, modules over a sufficiently friendly ring. Let $N\in\mathcal C$. We can consider the over-category $\mathcal C_{/N}$ of objects from $\mathcal C$ with a fixed morphism to $N$. Let $M\in\mathcal C_{/N}$ (the morphism $M\to N$ is implicitly understood). We can form the functor $$\begin{aligned} (-)\times_N M\colon \mathcal C_{/N} &\to \mathcal C_{/N},\\ X &\mapsto X\times_N M. \end{aligned}$$ Under which circumstances is this functor exact? I'd guess that in general it is only left exact (since I can write it as a kernel, and kernels are left exact), but I haven't worked out the details yet. Do you know an answer?