  The broadly applied notions of Lie bialgebras, Manin triples, classical
$r$-matrices and $\mathcal{O}$-operators of Lie algebras owe their importance
to the close relationship among them. Yet these notions and their
correspondences are mostly understood as classes of objects and maps among the
classes. To gain categorical insight, this paper introduces, for each of the
classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so
that the classes of objects become categories and the maps among the classes
become functors or category equivalences. For this purpose, we start with the
notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie
algebra endomorphism. We then generalize the above classical notions for Lie
algebras to endo Lie algebras. As a result, we obtain the notion of coherent
endomorphisms for each of the classes, which then generalizes to the notion of
coherent homomorphisms by a polarization process. The coherent homomorphisms
are compatible with the correspondences among the various constructions, as
well as with the category of pre-Lie algebras.
