With braiding With duals for objects category with duals (list of them) dualizable object (what they have) ribbon category, a.k.a. tortile category With duals for morphisms With traces Closed structure Special sorts of products Semisimplicity Morphisms Internal monoids Examples Theorems In higher category theory As every topos, a category of presheaves is cartesian closed monoidal. (cartesian closure of categories of presheaves) Let be a small category and write for its category of...