TITLE: Limits in the category of exact sequences
QUESTION [11 upvotes]: Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form
$$
0\to A\to B\to C\to 0
$$
and whose morphisms are triples of morphisms of $\mathbf C$ such that the diagram
$$
\begin{array}{ccccc}
A&\hookrightarrow & B & \to & C & \newline
\downarrow &&\downarrow && \downarrow \newline
A' &\hookrightarrow & B' &\to &  C'
\end{array}
$$
commutes in all its parts. Call this category $\boldsymbol\Sigma(\mathbf C)$.
How could one characterize the limits in $\boldsymbol\Sigma(\mathbf C)$? A little meditation shows that inverse systems are objects of the form
$$
\mathcal E_i\colon 0\rightarrow A_i\rightarrow B_i\rightarrow C_i\to 0
$$
(every object in the sequence can be thought as an element in a separate inverse system), so the universal property of $\varprojlim_\mathbf J \mathcal E_i$, whatever it turns out to be, must be enjoyed by
$$
\textstyle \varprojlim_\mathbf J \mathcal E_i : 0\to \varprojlim_\mathbf J A_i\to \varprojlim_\mathbf J B_i \to \varprojlim_\mathbf J C_i\to 0
$$
as soon as one looks to  ${A_i},{B_i},{C_i}$ as three inverse systems.
What condition(s?) has(ve?) to be imposed on them to make sure the sequence is exact?

REPLY [3 votes]: Considering the category $\Sigma_R(C) $ of the exact seqences like: $0\to A\to B\to C$ this category is complete and the inclusion $\Sigma(C)  \subset  \Sigma _R(C)$ is coreflexive (the coreflector comes from the coker of the last arrow), then limits in $\Sigma(C) $ exist and are given by the limit in $\Sigma_R(C) $ followed by the coreflection.
When the inclusion $\Sigma(C)  \subset  \Sigma _R(C)$ preserves (directed) limits is the content of Mittag-Leffler's theorem.