  We present a slick proof of completeness and cocompleteness for categories of
$F$-automata, where the span of maps $E\leftarrow E\otimes I \to O$ that
usually defines a deterministic automaton of input $I$ and output $O$ in a
monoidal category $(\mathcal K,\otimes)$ is replaced by a span $E\leftarrow F E
\to O$ for a generic endofunctor $F : \mathcal K\to \mathcal K$ of a generic
category $\mathcal K$: these automata exist in their `Mealy' and `Moore'
version and form categories $F\text{-}\mathsf{Mly}$ and
$F\text{-}\mathsf{Mre}$; such categories can be presented as strict 2-pullbacks
in $\mathsf{Cat}$ and whenever $F$ is a left adjoint, both
$F\text{-}\mathsf{Mly}$ and $F\text{-}\mathsf{Mre}$ admit all limits and
colimits that $\mathcal K$ admits. We mechanize some of of our main results
using the proof assistant Agda and the library `agda-categories`.
