If X is a type of structure that can be defined in a category, higher category, or category with some sort of structure, then the walking X is an informal term for the free category (resp. higher category, category with suitable structure) containing an X.
between the hom-set/category/space from to , for any , and the set/category/space of all Xs in .
The interval category is the walking arrow.
The syntactic category of a theory in some doctrine is the “walking -model” (in a -category). In particular, the classifying topos of a geometric theory is the “walking -model” qua Grothendieck topos (where the morphisms are the left-adjoint parts of geometric morphisms).