  Given Gray-categories $P$ and $L$, there is a Gray-category
$\mathrm{Tricat}_{\mathrm{ls}}(P,L)$ of locally strict trihomomorphisms with
domain $P$ and codomain $L$, tritransformations, trimodifications, and
perturbations. If the domain $P$ is small and the codomain $L$ is cocomplete,
we show that this Gray-category is isomorphic as a Gray-category to the
Gray-category $\mathrm{Ps}$-$T$-$\mathrm{Alg}$ of pseudo algebras, pseudo
functors, transformations, and modifications for a Gray-monad $T$ derived from
left Kan extension.
  Inspired by a similar situation in two-dimensional monad theory, we apply the
coherence theory of three-dimensional monad theory and prove that the the
inclusion of the functor category in the enriched sense into this Gray-category
of locally strict trihomomorphisms has a left adjoint such that the components
of the unit of the adjunction are internal biequivalences. This proves that any
locally strict trihomomorphism between Gray-categories with small domain and
cocomplete codomain is biequivalent to a Gray-functor. Moreover, the hom
Gray-adjunction gives an isomorphism of the hom 2-categories of
tritransformations between a locally strict trihomomorphism and a Gray-functor
with the corresponding hom 2-categories in the functor Gray-category. A notable
example is given by locally strict Gray-valued presheafs with small domain. Our
results have applications in three-dimensional descent theory and point into
the direction of a Yoneda lemma for tricategories.

Category Theory