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 $\mathcal{W}=Iso$)



Do limits in Waldhausen categories commute with ordinary limits?
Unless I'm misunderstanding your question, the answer is yes. Your second way of defining limits, via "the universal property of a diagram with some arrows in $\mathcal{F}$," is actually a special case of the normal definition of a limit, and limits commute. For your specific question of interest, the key observation is that $ker\_{\mathcal{F}}(f)$ is the pullback of the co-span below.


$$ \require{AMScd} \begin{CD}
ker\_{\mathcal{F}}(f) @>>> X
\\ @VVV @VVV
\\ 0 @>>> Y
\end{CD}
$$


where $0$ is the zero object.


