  It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched
category is meaningful for metric spaces: it captures exactly Cauchy-complete
metric spaces. In this paper we introduce the corresponding notion of Lawvere
completeness for $(\mathbb{T},\mathsf{V})$-categories and show that it has an
interesting meaning for topological spaces and quasi-uniform spaces: for the
former ones means weak sobriety while for the latter means Cauchy completeness.
Further, we show that $\mathsf{V}$ has a canonical
$(\mathbb{T},\mathsf{V})$-category structure which plays a key role: it is
Lawvere-complete under reasonable conditions on the setting; permits us to
define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$-categories.
