Homomorphisms and principal congruences of bounded lattices

Two years ago, I characterized the order $\text{Princl}L$ of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and $\text{gf}$ is a \zo homomorphism of $K$ into~$L$, then there is a natural isotone \zo-map ${\text{gf}}_{\text{Hom}}$ from $\text{Princl}K$ into $\text{Princl}L$. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map $\text{gy}$ of $P$ into $Q$, we represent $P$ and $Q$ as $\text{Princl}K$ and $\text{Princl}L$ for bounded lattices $K$ and $L$ with a \zo homomorphism $\text{gf}$ of $K$ into $L$, so that $\text{gy}$ is represented as ${\text{gf}}_{\text{Hom}}$.