Let $\mathcal A$ be an abelian category, $\{X_i,f_{ij}\}_{i\leqslant j\in I}$ a direct system of $\mathcal A$ such that for any $i\leqslant j\in I$, $f_{ij}:X_i\to X_j$ is an monomorphism.
Suppose the colimit of $\{X_i,f_{ij}\}$ exists, can we deduce that the structural morphisms $u_i:X_i\to \varinjlim X_i$ are monomorphisms?
What I know is that it holds when $\mathcal A$ has enough injective objects and $I=\mathbb N$.
Not necessarily. For instance, let $\mathcal{A}$ be the category of finite abelian groups and take the direct system $$\mathbb{Z}/2\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}/4\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}/8\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}/16\mathbb{Z}\stackrel{2}{\to}\mathbb{Z}/32\mathbb{Z}\stackrel{2}{\to}\dots$$
Given a cocone from this diagram to a finite abelian group $A$, note that every element of $\mathbb{Z}/2^n\mathbb{Z}$ must map to an element of $A$ that is both $2^n$-torsion and infinitely divisible by $2$. But in a finite abelian group, such an element must be $0$. It follows that the trivial group $0$ is a colimit of this system in $\mathcal{A}$.
In fact, this is not even true assuming $\mathcal{A}$ has enough injectives. For instance, take $\mathcal{A}=Ab^{op}$: there exists an inverse system of surjections of nontrivial abelian groups such that the inverse limit is trivial (see, for instance, Section 4 of https://math.berkeley.edu/~gbergman/papers/unpub/emptylim.pdf). Explicitly, to get such an inverse system, you can take an inverse system of surjections of nonempty sets indexed by $\omega_1$ with empty inverse limit, and apply the free abelian group functor.