The Stacks project

Example 19.2.2. Next we give an example where the map fails to be injective. Let $B_ n = \mathbf{N}/\{ 1, 2, \ldots , n\} $, that is, the quotient set of $\mathbf{N}$ with the first $n$ elements collapsed to one element. There are natural maps $B_ n \to B_ m$ for $n \leq m$, so the $\{ B_ n\} $ form a system of sets over $\mathbf{N}$. It is easy to see that $\mathop{\mathrm{colim}}\nolimits B_ n = \{ *\} $: it is the one-point set. So it follows that $\mathop{\mathrm{Mor}}\nolimits (A, \mathop{\mathrm{colim}}\nolimits B_ n)$ is a one-element set for every set $A$. However, $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits (A , B_ n)$ is not a one-element set. Consider the family of maps $A \to B_ n$ which are just the natural projections $\mathbf{N} \to \mathbf{N}/\{ 1, 2, \ldots , n\} $ and the family of maps $A \to B_ n$ which map the whole of $A$ to the class of $1$. These two families of maps are distinct at each step and thus are distinct in $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits (A, B_ n)$, but they induce the same map $A \to \mathop{\mathrm{colim}}\nolimits B_ n$.