New concept: metamonovalued morphisms

Let fix some dagger category every of Hom-sets of which is a complete lattice, and the dagger functor agrees with the lattice order.

I define a morphism $latex f$ to be monovalued when $latex \circ f^{-1}\le \mathrm{id}_{\mathrm{Dst}\, f}$.

I call a morphism $latex f$ metamonovalued when $latex (\bigwedge G) \circ f = \bigwedge_{g \in G} ( g \circ f)$ for every set $latex G$ of morphims (provided that the sources and domains of the morphisms are suitable). Here $latex \bigwedge$ denoted the infimum on the above mentioned complete lattice.

In my book (recently added to the preprint) I have proved that every monovalued funcoid is metamonovalued. Thus there is probably some connection between monovalued and metamonovalued morphisms.

The following problems are yet open:

Question Is every metamonovalued funcoid monovalued?

Question Is every metamonovalued reloid monovalued?

Question Is every monovalued reloid metamonovalued?