Counter-example against “inward reloid of corresponding funcoid of a convex reloid” conjecture

I found (a rather trivial) counter-example to one conjecture I considered in the past:

Example (\mathsf{RLD})_{\mathrm{in}} (\mathsf{FCD}) f \neq f for some convex reloid f.

Proof Let f = (=). Then (\mathsf{FCD}) f = (=). Let a is some nontrivial atomic filter object. Then (\mathsf{RLD})_{\mathrm{in}} (\mathsf{FCD}) (=) \supseteq a \times^{\mathsf{FCD}} a \nsubseteq (=) and thus (\mathsf{RLD})_{\mathrm{in}} (\mathsf{FCD}) (=) \nsubseteq (=).

Before I found the last counter-example, I was thinking that (\mathsf{FCD}) is an isomorphism from the set of of funcoids to the set of convex reloids. As this conjecture failed, we need an other way to characterize the set of reloids isomorphic to funcoids.

The first thing we need to check is whether (\mathsf{RLD})_{\mathrm{in}} is an injection. Maybe this is a simple problem (or maybe it is hard) but I haven’t thought about it yet.

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