Limits of tight acset morphisms

Currently, tight acset morphisms don't support limits. But most limits in this category are actually easy: connected limits (notably equalizers) in slice categories are computed as in the underlying category, and products are pullbacks. So we can compute equalizers of tight acset morphisms objectwise over $S_0$ and binary products by pulling back over weights, i.e. $(F\times G)(s:S_0)=\{(f,g)\in F(s)\times g(s)\mid \forall w : S_\to(s,\top) w(f)=w(g)\}.$ (Sorry, curly braces aren't escaping right?)

Terminal objects won't be finite when attribute types aren't, so that's no good, but it seems like it might be nice to have everything else?