  We prove a biadjoint triangle theorem and its strict version, which are
$2$-dimensional analogues of the adjoint triangle theorem of Dubuc. Similarly
to the $1$-dimensional case, we demonstrate how we can apply our results to get
the pseudomonadicity characterization (due to Le Creurer, Marmolejo and
Vitale).
  Furthermore, we study applications of our main theorems in the context of the
$2$-monadic approach to coherence. As a direct consequence of our strict
biadjoint triangle theorem, we give the construction (due to Lack) of the left
$2$-adjoint to the inclusion of the strict algebras into the pseudoalgebras.
  In the last section, we give two brief applications on lifting biadjunctions
and pseudo-Kan extensions.

Category Theory