I'm reading about spectral sequences from various places (e.g. "The heart of cohomology" by Goro Kato, the Stack Project, Wikipedia), and I have a doubt.
We consider a bigraded cohomological spectral sequence ($E_r^{p,q},\mathrm{d}_r^{p,q}$) in an abelian category $\mathcal{A}$, with $\mathrm{d}_r^{p,q}:E_r^{p,q}\rightarrow E_r^{p+r,q-r+1}$. We can define the objects $B^sE_r^{p,q}$ and $Z^{s}E_r^{p,q}$, in such a way that we have
$ 0=B^0E_r^{p,q}\subseteq B^1E_r^{p,q}\subseteq\dots\subseteq B^sE_r^{p,q}\subseteq\dots\subseteq Z^sE_r^{p,q}\subseteq\dots\subseteq Z^1E_r^{p,q}\subseteq Z^0E_r^{p,q}=E_r^{p,q}. $
Now, if they exist (for example if $\mathcal{A}$ ha countable products and coproducts), we can define the objects $B^{\infty}E_r^{p,q}=\mathrm{colim}B^sE_r^{p,q}$ and $Z^{\infty}E_r^{p,q}=\mathrm{lim}Z^sE_r^{p,q}$. These are filtered co/limits, and they are usually written as $B^{\infty}E_r^{p,q}=\bigcup B^sE_r^{p,q}$ and $Z^{\infty}E_r^{p,q}=\bigcap Z^sE_r^{p,q}$, since they indeed have this form in the case of $\mathcal{A}$ being a category of modules or other similar "concrete" categories.
By the various universal properties involved, we have a morphism $j_r^{p,q}:B^{\infty}E_r^{p,q}\rightarrow Z^{\infty}E_r^{p,q}$, and we can then set $E_{\infty}^{p,q}=\mathrm{Coker}j_r^{p,q}$ (one should then verify that this is "independent on r", i.e. all these objects are isomorphic).
The authors, though, write this as $E_{\infty}^{p,q}=\frac{Z^{\infty}E_r^{p,q}}{B^{\infty}E_r^{p,q}}$, and so they are implicitly assuming that $j_r^{p,q}$ is injective. Though I can clearly see this is true in some concrete examples (e.g. if $\mathcal{A}$ is a category of modules), I don't see if and why this is true in general, since in general taking colimits is not an exact operation.
So, my question is, is it always true that $j_r^{p,q}:B^{\infty}E_r^{p,q}\rightarrow Z^{\infty}E_r^{p,q}$ is always injective?