  There is a well-known correspondence between coherent theories (and their
interpretations) and coherent categories (resp. functors), hence the
(2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent
functors and all natural isomorphisms) is of logical interest. We prove that
this category admits all small 2-limits and 2-colimits (in the ($\infty
$,1)-categorical sense), and prove a 2-categorical small object argument to
provide weak factorisation systems for coherent functors.
