Less than a hour ago I discovered a new mapping from funcoids to reloids:

**Definition** $latex (\mathsf{RLD})_X f = \bigsqcap \left\{ g \in \mathsf{RLD} \mid (\mathsf{FCD}) g \sqsupseteq f \right\}$ for every funcoid $latex f$.

Now I am going to work on the following conjectures:

**Conjecture** $latex (\mathsf{RLD})_X f = \min \left\{ g \in \mathsf{RLD} \mid (\mathsf{FCD}) g \sqsupseteq f \right\}$, that is $latex (\mathsf{RLD})_X $ is the lower adjoint of $latex (\mathsf{FCD})$.

**Conjecture** $latex (\mathsf{RLD})_X f = f$ if $latex f$ is a principal funcoid.

**Conjecture** $latex (\mathsf{RLD})_X (f|_\mathcal{A}) = ((\mathsf{RLD})_X f)|_\mathcal{A}$.

Note that from the two last conjectures it follows that $latex (\mathsf{RLD})_X \mathrm{id}^{\mathsf{FCD}}_\mathcal{A} = \mathrm{id}^{\mathsf{RLD}}_\mathcal{A}$.

$latex (\mathsf{RLD})_X$ is

nota lower adjoint of $latex (\mathsf{FCD})$ because $latex (\mathsf{FCD})$ does not preserve binary meets.