I'm working in the category $C$ of sets of (possibly but not necessarily infinite) binary sequences. Let $E_n$ be the object of $C$ consisting of sequences of all length $n$. I consider the diagram in $C$ $$ E_1\xrightarrow{f_1}E_2\xrightarrow{f_2}E_4\xrightarrow{f_3}\cdots\xrightarrow{f_i}E_{2^i}\xrightarrow{f_{i+1}} \cdots$$ The arrows consist of sending the element $x\in E_{2^i}$ to the element in $E_{2^{i+1}}$ with first $2^i$ elements the same as $x$ and last $2^i$ just $0$s. So $(0,1)$ maps to $(0, 1, 0, 0 )$. I would like to know what the colimit (or direct limit) of this diagram is, I think is a matter of applying definitions. I am close to 100% sure that it is the set consisting of binary sequences that are eventually zero and NOT the set of all binary sequences.
I should explain the background here, though this is unnecessary for the problem above. I am studying the Kan fibrant replacement functor $Ex^\infty$ in simplicial set theory which is defined in a similar way to the above. In particular I was using without proof the "fact" that $Ex^\infty (Ex^\infty X)=Ex^\infty(X)$. This feels like it should be true (subdivision of a subdivision is still subdivision), but if I'm right about the above the map $Ex^\infty(X)\to Ex(Ex^\infty(X))$ is not an isomorphism. This is something so fundamental I feel like I need a second pair of eyes to look at it, which is not possible at the minute due to the virus. Thanks a million in advance for your help!
This sequence is naturally isomorphic to another sequence of sets. Namely let $S_{2^i} \approx E_{2^i}$ be the set of all infinite binary sequences that are zero after the $2^i$th number. Then letting $S_{2^i} \rightarrow S_{2^{i+1}}$ be the inclusion we see that the isomorphism $S_* \rightarrow E_*$ commutes with the relevant maps and so is a natural isomorpism.
This implies $\text{colim}(S_*) = \text{colim}(E_*)$, but the colimit of $S_*$ is just the union of all the $S_{2^i}$ since $S_{2^i} \rightarrow S_{2^{i+1}}$ is just the inclusion of subsets.
Then we see that an infinite sequence of zeros and ones $(x_j)_j$ is in $\text{colim}(S_*)$ iff $(x_j)_j \in S_{2^{i}}$ for some $i$ by definition of union so $x_j$ has to be zero eventually and can't be any infinite sequence.