Two directed colimits of same spaces with different inclusions For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets. 
Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \} $$
and  $$Y=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{j_n} X_{n+1}\cdots \}$$

Question: I'm looking for an example where $X$ is not isomorphic to
  $Y$ as a simplicial set.

 A: Collecting the comments, we have the following two observations:


*

*Consider the functor $NB: Grp \to Cat \to sSet$, where the first functor,B, takes a group to the category with one object and endomorphisms equal to the group, and the second,N, is the nerve. Note that:


A) B factors through Cat_1, the categories with one object. We have a right adjoint to B given by the automorphism group of the only object involved. Thus $B$ is a left adjoint composed with a full inclusion $Cat_1 \to Cat$. The latter evidently preserves colimits. Moreover, B takes injectives to maps that are injective on both objects and arrows.
B) The nerve is given by the formula $NC_n =Hom([n], C)$. Note that the objects $[n]$ are finite, so that morphisms from $[n]$ to a directed colimit $\{X_k\}$ indeed end up on one of the objects $X_k$. A rephrasing of this gives exactly that N commutes with directed colimits. Also, it is evident from the formula that N takes injective on both objects and arrows to injective levelwise, which are the mono of sSets.
On balance, we can reduce to find an example in Groups.


*Take, as wisely suggested by Todd Trimble, the two sequences
$$ \mathbb{Z} = ... = \mathbb{Z} = ... $$
$$ \mathbb{Z} \overset{2*}{\to} ... \mathbb{Z} \overset{2*}{\to}... $$
The first sequence evidently has colimit Z. The other one is not: indeed, we show that there are no non-zero morphisms from the second colimit, W, to Z modulo 2. Suppose there is one collection $\{f_k\}$ with one of them non zero. Say $f_j(x)=1$ for some j,x. Then 
$$0= 2f_{j+1}(x) = f_{j+1}(2x) = f_j(x) = 1$$
A contradiction.
