Limits in the category of exact sequences

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?

Solution 1:

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.