  Let $\mathsf{Q}$ be a commutative and unital quantale. By a $\mathsf{Q}$-map
we mean a left adjoint in the quantaloid of sets and $\mathsf{Q}$-relations,
and by a partial $\mathsf{Q}$-map we refer to a Kleisli morphism with respect
to the maybe monad on the category $\mathsf{Q}\text{-}\mathbf{Map}$ of sets and
$\mathsf{Q}$-maps. It is shown that every $\mathsf{Q}$-map is symmetric if and
only if $\mathsf{Q}$ is weakly lean, and that every $\mathsf{Q}$-map is exactly
a map in $\mathbf{Set}$ if and only $\mathsf{Q}$ is lean. Moreover, assuming
the axiom of choice, it is shown that the category of sets and partial
$\mathsf{Q}$-maps is monadic over $\mathsf{Q}\text{-}\mathbf{Map}$.
