Just a few seconds ago I had an idea how to generalize both funcoids and reloids.

Consider

• a precategory, whose objects are sets
• product $\times$ of filters on sets ranging in morphisms of this category
• operations $\mathrm{dom}$ and $\mathrm{im}$ from the morphisms of our precategory to filters on our objects (sets)

This axiomatic system is so powerful that it allows to define $\langle f\rangle$ for a funcoid $f$:

$\langle f\rangle\mathcal{X} = \mathrm{im}(f\circ(1^\mathfrak{F}\times^\mathsf{FCD}\mathcal{X}))$.

However this axiomatic system is probably too weak to prove $\langle g\rangle\langle f\rangle\mathcal{X} = \langle g\circ f\rangle\mathcal{X}$. We need additional axioms.