Disclaimer : I asked this question on MSE, I have no answer and I think it's better to ask it here.

Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category. On one hand, we can define the ordinary limits $lim_A$ of the underlying category $A$. On other hand, we can define limits of Waldhausen categories via the universal property of a diagram with some arrows in $\mathcal{C}$ . For example we can define $ker_{\mathcal{C}}(f) \stackrel{i}{\rightarrow}X\stackrel{f}{\rightarrow}Y$ where $i \in \mathcal{C}$ has the universal property of the kernel for $j \in \mathcal{C} | fj=0$.

My question is: if they exist, do ordinary limits and Waldhausen limits commute? In particular, do ordinary countable products and Waldhausen kernels commute?

Or : do we have some conditions such that ordinary limits and Waldhausen limits commute?

(I'm interested by the second case, but the first one implies the second one. And in my particular case $\mathcal{W}=Iso$)