  This is the introductory chapter of my PhD Thesis. This thesis consists of
one introductory chapter and four single-authored papers written during my PhD
studies at the University of Coimbra under supervision of Maria Manuel
Clementino. In this first chapter, we give a glance of the scope of our work
and briefly describe elements of the original contributions of each paper,
including some connections between them. We also give a brief exposition of our
main setting, which is $2$-dimensional category theory. In this direction: (1)
we give an exposition on the doctrinal adjunction, focusing on the
Beck-Chevalley condition as used in Chapter "Pseudo-Kan Extensions and Descent"
(arXiv: 1606.04999), (2) we apply the results of "On lifting of biadjoints and
lax algebras" (arXiv: 1607.03087) in a generalized setting of the formal theory
of monads and (3) we apply the biadjoint triangle theorem of "On biadjoint
triangles" (aXiv: 1606.05009) to study (pseudo)exponentiable pseudocoalgebras.
