Direct limit of a subdirected system

Question

Let $(M_{i},\phi_{ij})$ be a direct system over a directed set $I$ and let $M$ be its direct limit. Suppose we remove a finite/infinite number of terms from the direct system $(M_{i},\phi_{ij})$. Does this/can this change the limit?

Answer 1

(There was a comment from the op clarifying that he was asking if it's possible to change the limit by changing $I$. This answer is in response to that)

To expand, if we even just limit ourselves to sets this is likely we have changed the limit.

If we take any suitably large universe set, $U\in\operatorname{Ob}(\mathbf{Sets})$, and then set $U=I$ to be our indexing set and declare $U_x=\{x\}$ for $x\in U$, then the direct limit is merely $U$, when we have all $\phi_{ii}$ the identities, and no other $\phi_{ij}$. If we remove anything from $I$, say $\{U_y\}_{y\in J}$ then the limit becomes $U\setminus J\ne U$.

Edit To keep all the relevant stuff together, I'll respond to the follow-up comment from the main post here as well:

There's no real way to fix this with a simple condition on just $I$. If you take $I=\{1,2,3\}$ and $U_1=\{1,2\}, U_2=\{2, 3\}, U_3=\{3\}$ then you can remove either $U_2$ or $U_3$ and preserve the set direct limit (i.e. union) but not much else can be said just purely in terms of $I$ without looking at the actual objects indexed by it. It's not a matter of the indexing set nearly so much as the the actual relations involved. You can say something in the case you have a monic in a concrete category, when this reduces to an injection, so you can safely omit the sub-object without changing the limit. (This assumes you have one $\phi_{ij}$ as the inclusion of the sub-object). Note, however, this still relies directly on the actual relationships, and not just the indexing set, $I$.