  Building on our previous work on enriched universal algebra, we define a
notion of enriched language consisting of function and relation symbols whose
arities are objects of the base of enrichment. In this context, we construct
atomic formulas and define the regular fragment of enriched logic by taking
conjunctions and existential quantifications of those. We then characterize
enriched categories of models of regular theories as enriched injectivity
classes in the enriched category of structures. These notions rely on the
choice of a factorization system on the base of enrichment which will be used
to interpret relation symbols and existential quantifications.
