  Lawvere observed in his celebrated work on hyperdoctrines that the
set-theoretic schema of comprehension can be elegantly expressed in the
functorial language of categorical logic, as a comprehension structure on the
functor $p:\mathscr{E}\to\mathscr{B}$ defining the hyperdoctrine.
  In this paper, we formulate and study a strictly ordered hierarchy of three
notions of comprehension structure on a given functor
$p:\mathscr{E}\to\mathscr{B}$, which we call (i) comprehension structure, (ii)
comprehension structure with section, and (iii) comprehension structure with
image.
  Our approach is 2-categorical and we thus formulate the three levels of
comprehension structure on a general morphism
$p:\mathrm{\mathbf{E}}\to\mathrm{\mathbf{B}}$ in a 2-category $\mathscr{K}$.
  This conceptual point of view on comprehension structures enables us to
revisit the work by Fumex, Ghani and Johann on the duality between
comprehension structures and quotient structures on a given functor
$p:\mathscr{E}\to\mathscr{B}$.
  In particular, we show how to lift the comprehension and quotient structures
on a functor $p:\mathscr{E}\to\mathscr{B}$ to the categories of algebras or
coalgebras associated to functors $F_{\mathscr{E}}:\mathscr{E}\to\mathscr{E}$
and $F_{\mathscr{B}}:\mathscr{B}\to\mathscr{B}$ of interest, in order to
interpret reasoning by induction and coinduction in the traditional language of
categorical logic, formulated in an appropriate 2-categorical way.
