  Composing with the inclusion $\mathsf{Set}\to\mathsf{Cat} $, a graph $G$
internal to $\mathsf{Set} $ becomes a graph of discrete categories, the
coinserter of which is the category freely generated by $G$. Introducing a
suitable definition of $n$-computad, we show that a similar approach gives the
$n$-category freely generated by an $n$-computad. Suitable $n$-categories with
relations on $n$-cells are presented by these $(n+1)$-computads, which allows
us to prove results on presentations of thin groupoids and thin categories. So
motivated, we introduce a notion of deficiency of (a presentation of) a
groupoid via computads and prove that every small connected thin groupoid has
deficiency $0$. We compare the resulting notions of deficiency and presentation
with those induced by monads. In particular, we find our notion of group
deficiency to coincide with the classical one. Finally, we study presentations
of $2$-categories via $3$-computads, focusing on locally thin groupoidal
$2$-categories. Under suitable hypotheses, we give efficient presentations of
some locally thin and groupoidal $2$-categories. A fundamental tool is a
$2$-dimensional analogue of the association of a "topological graph" to every
graph internal to $\mathsf{Set} $. Concretely, we construct a left adjoint
$\mathcal{F}_ {\mathsf{Top} _ 2} : 2\textrm{-}\mathsf{cmp}\to \mathsf{Top} $
associating a $2$-dimensional CW-complex to each small $2$-computad. Given a
$2$-computad $\mathfrak{g} $, the groupoid it presents is equivalent to the
fundamental groupoid of $\mathcal{F} _ {\mathsf{Top} _2}(\mathfrak{g}) $.
Finally, we sketch the $3$-dimensional version $\mathcal{F}_ {\mathsf{Top} _
3}$.
