I am working through the definition of the restriction of a sheaf to a subspace, and proving that if $Z \subset X$, and $\mathscr{F}$ is a sheaf on $X$, then $(\mathscr{F}|_Z)_p = \mathscr{F}_p$ for any point $p \in Z$. From the definitions in Hartshorne, this reduces to proving that
$$\varinjlim_{V \ni p}\left(\varinjlim_{U \supset V} \mathscr{F}(U)\right) = \varinjlim_{U \ni p} \mathscr{F}(U).$$
I can work through the argument using the universal property of direct limits, since if $p \in U \subset V$, then the diagram of open sets containing $V$ is contained in the diagram of open sets containing $U$. My argument is the same as alluded to here. I don't know much category theory though, and I'm wondering whether there is a cleaner, more categorical way to see when taking two colimits can be reduced to taking one colimit?