Consider the map \(\mathbb{Z} \xrightarrow{3z} \mathbb{R}\) which sends an integer to \(3z\) in the reals.
To find a left adjoint for this map, we write \(\lceil r \rceil\) for the smallest natural above \(r \in \mathbb{R}\) and \(\lfloor r \rfloor\) for the largest integer below \(r \in \mathbb{R}\)
The left adjoint is \(\lceil r/3 \rceil\)
Check: \(\lceil x/3 \rceil \leq y\) \(\iff x \leq 3y\)