On the canonical algebraic structure of a category

For any locally small category image, applying Lawvereʹs “structure” functor to the hom-functor image produces a Lawvere theory image, called the canonical algebraic structure of image, and given by image — provided that this latter set is small, which is certainly the case when image is complete and cocomplete and some small subset of its objects is either generating or co-generating. If now image is a commutative theory, so that image-Alg is a symmetric monoidal closed category, enrichments of image over image-Alg correspond to liftings of H through the forgetful functor image-Alg→Set, and hence, (by Lawvereʹs structure-semantics adjunction) to theory-maps image. In fact, whenever image admits either finite powers or finite multiples, the theory image is itself commutative, so that image has a canonical enrichment over image. When image is of the form image-Alg for some theory image we find that image, each being isomorphic to the centre of image. We end by considering the situation where image is already enriched over some symmetric monoidal category image, and may in particular be image itself.