  In a category $\mathcal{C}$ with a proper $(\mathcal{E},
\mathcal{M})$-factorization system, we study the notions of strict, co-strict,
initial and final morphisms with respect to a topogenous order. Besides showing
that they allow simultaneous study of four classes of morphisms obtained
separately with respect to closure, interior and neighbourhood operators, the
initial and final morphisms lead us to the study of topogenous structures
induced by pointed and co-pointed endofunctors. We also lift the topogenous
structures along an $\mathcal{M}$-fibration. This permits one to obtain the
lifting of interior and neighbourhood operators along an
$\mathcal{M}$-fibration and includes the lifting of closure operators found in
the literature. A number of examples presented at the end of the paper
demonstrates our results.
