I'm learning Hartshorne's Algebraic Geometry and trying to solve the following exercise: Exercise II.1.4
Let $\phi: \mathcal{F} \rightarrow \mathcal{G}$ be a morphism of presheaves such that $\phi(U): \mathcal{F}(U) \rightarrow \mathcal{G}(U)$ is injective for each $U$. Show that the induced map (on the sheafification) $\phi^{+}: \mathcal{F}^{+} \rightarrow \mathcal{G}^{+}$ is injective.
Note that there are bijections between the stalks $\mathcal{F}^{+}_p$ of the sheafification sheaf $\mathcal{F}^{+}$ and stalks $\mathcal{F}_p$ of the presheaf $\mathcal{F}$ for any $p \in X$, it suffices to show that $\phi$ is injective on stalks of the presheaves, i.e. $\phi_p : \mathcal{F}_p \rightarrow \mathcal{G}_p $ is injective for all $p \in X$. This can be proved by direct verification.
Here I hope to generalize the discussion, and my question is how to prove (or disprove):
Proposition: Let $I$ be a filtered preordered set, fix inductive systems $((X_i)_i, (\alpha_{ji})_{i \leq j})$ and $((Y_i)_i, (\beta_{ji})_{i \leq j})$ of sets, indexed by $I$, and let $X$ and $Y$ be their inductive limits. Let $(u_i: X_i \rightarrow Y_i)_i$ be a morphism of inductive systems and let $u:X \rightarrow Y$ be its inductive limit. Show that if there exists an $i \in I$ such that $u_j$ is injective (resp. surjective) for all $j \geq i$, then $u$ is injective (resp. surjective).
Since I worked out the Hartshorne's exercise by discussing the concrete stalks and germs instead of thinking on the "categorial" level, I find it hard to verify the above "Proposition". I'm also not familiar with the inductive limit of morphisms of inductive systems.
Could anyone prove the above proposition in details, or provide some references on these?
Moreover, does similar proposition holds for projective limits?
Thank you for your helps! :)
P.S. The proposition is adapted from Ulrich Gortz, Torsten Wedhorn's Algebraic Geometry 1 - Schemes.