I know that it is not true that for a map $f \colon X \rightarrow Y$ of schemes,
injectivity (on underlying sets) of $f$ gives a monomorphism in the category
of schemes.  

Stronger assumptions are required, see e.g. [Tag 01L6](https://stacks.math.columbia.edu/tag/01L6) in the Stacks project.

But I have not come up with a counterxample yet.
So can someone provide me with some concrete counterexample,
preferrably some well-known class of such morphisms?