Labelled hypergraph copy count

For hypergraphs $F,H$, a labelled copy is an injective function $\varphi :V(F)\to V(H)$ taking every hypergraph edge of $F$ to an hypergraph edge of $H$. Specified vertex of a hypergraph classes may additionally require $\varphi (i)\in V_i$. The count is the sum, over these injective maps, of the product of the host hypergraph edge indicators. Without injectivity the same expression counts hypergraph homomorphisms. On a common host of size $n$, their counts differ by at most $\left(\genfrac{}{}{0ex}{}{|V(F)|}{2}\right)n^{|V(F)|-1}$: every noninjective map identifies at least one pair of abstract vertices of a hypergraph, each prescribed equality permits at most $n^{|V(F)|-1}$ maps, and the union bound completes the estimate. An induced copy additionally preserves nonedges, so its indicator product includes nonedge factors. The counting lemma for octahedrally quasirandom three-uniform hypergraphs estimates these labelled products when the requisite centered triple indicators have small box norm.