Homomorphisms and principal congruences of bounded lattices

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

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