Suppose we have a ringed space $(X, \mathcal{O}_{X})$ with sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$. Let $I$ be an indexing set and suppose we have a collection of subsets $U_{i} \subseteq X$ along with a collection of sections $s_{i} \in \mathcal{F}(U_{i})$ both indexed by $I$. The subsheaf generated by the family $\{ s_{i} \}_{i \in I}$ is defined to be the image of the canonical morphism $$ \phi: \bigoplus_{i} \mathcal{O}_{U_{i}} \longrightarrow \mathcal{F}, $$ where $\mathcal{O}_{U_{i}}$ denotes the extension by zero of the restriction to $U_{i}$.
Why is the collection of all finitely generated subsheaves a directed system?
I feel like this is a stupid question, because it seems so blatantly intuitively obvious that if $J \subseteq I$ is a subset (where both $J$ and $I$ are finite), then the subsheaf generated by the set $J$ will be a subsheaf of the subsheaf generated by the set $I$, but I can't see how to prove this using the formalism above.
In particular, if I have \begin{align} \phi: \bigoplus_{i} \mathcal{O}_{U_{i}} \longrightarrow \mathcal{F} \\ \psi: \bigoplus_{j} \mathcal{O}_{U_{j}} \longrightarrow \mathcal{F} \end{align} how do I construct a monomorphism $\text{im} (\psi) \rightarrow \text{im}(\phi)$? Is there another way to see immediately that the resulting collection is actually a directed system?