  In this paper we carry the construction of equilogical spaces into an
arbitrary category $\mathsf{X}$ topological over $\mathsf{Set}$, introducing
the category $\mathsf{X}$-$\mathsf{Equ}$ of equilogical objects. Similar to
what is done for the category $\mathsf{Top}$ of topological spaces and
continuous functions, we study some features of the new category as
(co)completeness and regular (co-)well-poweredness, as well as the fact that,
under some conditions, it is a quasitopos. We achieve these various properties
of the category $\mathsf{X}$-$\mathsf{Equ}$ by representing it as a category of
partial equilogical objects, as a reflective subcategory of the exact
completion $\mathsf{X}_{_{{\rm ex}}}$, and as the regular completion
$\mathsf{X}_{_{{\rm reg}}}$. We finish with examples in the particular cases,
amongst others, of ordered, metric, and approach spaces, which can all be
described using the $(\mathbb{T},\mathsf{V})$-$\mathsf{Cat}$ setting.
