Pullback along a direct system

Question

Let $\mathcal{A}$ be a Grothendieck category. Consider a short exact sequence in $\mathcal{A}$: $$ 0 \to A \to B\to C \to 0$$ and let $C = \varinjlim C_i$, direct limit of a direct system of objects. Let $B_i$ be the pullback of the maps $B \to C$ and $C_i \to C$, for any $i$. Is it possible to define a direct system $(B_i, \phi_{ij})$, with $\phi_{ij} \colon B_i \to B_j$, in such a way $B = \varinjlim B_i$?