The Stacks project

Example 19.2.1. Consider the category of sets. Let $A = \mathbf{N}$ and $B_ n = \{ 1, \ldots , n\} $ be the inductive system indexed by the natural numbers where $B_ n \to B_ m$ for $n \leq m$ is the obvious map. Then $\mathop{\mathrm{colim}}\nolimits B_ n = \mathbf{N}$, so there is a map $A \to \mathop{\mathrm{colim}}\nolimits B_ n$, which does not factor as $A \to B_ m$ for any $m$. Consequently, $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits (A, B_ n) \to \mathop{\mathrm{Mor}}\nolimits (A, \mathop{\mathrm{colim}}\nolimits B_ n)$ is not surjective.