While walking home from McDonalds I conceived the following idea how we can generalize reloids and funcoids.

Let $C$ be a category with finite products, the set of objects of which is a complete lattice (for the case of funcoids as described below it is enough to be just join-semilattice).

One can argue which axioms $C$ shall obey. For example, it is yet unclear, whether we should require the product to be distributive over any suprema or just over finite suprema.

By definition, generalized relation is a supremum of some set of binary products of objects of our category.

We can define generalized reloids as filters on generalized relations.

Similarly we can define generalized funcoids replacing arbitrary suprema in the definition of generalized relation with finite suprema. (See this online article for a proof that in the case $C=\mathbf{Set}$ this is equivalent with funcoids.)

Note, that I am going to do this research in the second volume of my research monograph, having said that the first volume is yet in process of rewriting and is not yet published.

I’ve corrected the post: The set of objects of the category should be a complete lattice, not the Hom-sets. And the example category is $\mathbf{Set}$ not $\mathbf{Rel}$.
An empty meet does not exist in $\mathbf{Set}$. We need to work around this “feature”.