Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.

Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of short exact sequences.

Is it true to say that $\varepsilon: 0\to \lim A_i \to \lim B_i \to \lim C_i \to 0$ (where $lim$ denotes the direct limit) is the direct limit $\lim \varepsilon_i$.

Actually we know that there exists a morphism $\varepsilon_i\to \varepsilon$ for each $i$.

What does occur when $\varepsilon_i$ is a split short exact sequence for each $i$? In fact why in this fact the direct limit is a pure short exact sequence?

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.

Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of short exact sequences.

Is it true to say that $\varepsilon: 0\to \lim A_i \to \lim B_i \to \lim C_i \to 0$ (where $lim$ denotes the direct limit) is the direct limit $\lim \varepsilon_i$.

Actually we know that there exists a morphism $\varepsilon_i\to \varepsilon$ for each $i$.

What does occur when $\varepsilon_i$ is a split short exact sequence for each $i$? In fact why in this fact the direct limit is a pure short exact sequence?

In a locally finitely presented category, an exact sequence is called pure if it is a direct limit of split short exact sequences. In a locally presentable category the notion of $\alpha$-purity arises in a similar way (In fact pure exact sequences are $\aleph$-pure exact sequences).

For example in the category of modules over a ring, an exact sequence is pure if and only if it is a direct limit of split exact sequences.