An object \(E\) in a 2-category equipped with a Yoneda structure was defined [R. Street and R. Walters, J. Algebra 50, 350–379 (1978; Zbl 0401.18004)] to be totally cocomplete (abbreviated to “total”) when the Yoneda morphism \(y_E : E\to \mathcal{P}E\) had a left adjoint \(z_E\dashv y_E\). In particular this applies in the 2-category of categories with homs enriched in a nice (say locally finitely presentable) monoidal category \(\mathscr{V}\). When the morphisms in the 2-category are restricted to finite limit preserving \(\mathscr{V}\)-functors, we call a total object \(E\) lex total (“lex” for “left exact). When \(\mathscr{V} = \mathrm{Set}\) and \(E\) is not too big [R. Street, Bull. Aust. Math. Soc. 23, 199–208 (1981; Zbl 0471.18005)], \(E\) is lex total if and only if it is a Grothendieck topos. In the absence of such an existence theorem for a generating set for general \(\mathscr{V}\), it makes sense to define a \(\mathscr{V}\)-topos to be a lex-total \(\mathscr{V}\)-category with a small strongly generating set of objects; indeed, in [B. Day and R. Street, Contemp. Math. 431, 187–202 (2007; Zbl 1128.18004)] we also defined monoidal \(\mathscr{V}\)-topos. A feature of a Grothendieck topos is that any generating set of its objects is automatically dense (or “left adequate” in Isbell’s terminology); this is a key ingredient of the Giraud Theorem characterizing Grothendieck toposes as it is in the Gabriel-Popescu Theorem characterizing Grothendieck abelian categories. In [B. Day and R. Street, J. Pure Appl. Algebra 43, 235–242 (1986; Zbl 0612.18004)] we saw for general \(\mathscr{V}\) that this feature has to do with the left adjoint \(z_E\) preserving monomorphisms. When \(\mathscr{V}\) is the category \(\mathrm{Ab}\) of abelian groups (so that left adjoints preserving monomorphisms are automatically finite limit preserving and all epimorphisms are strong), a \(\mathscr{V}\)-topos is precisely a Grothendieck abelian category.
A Grothendieck \(\mathscr{V}\)-enriched category in the present paper is a \(\mathscr{V}\)-topos as above. When \(\mathscr{V}\) itself is a “Grothendieck cosmos” (which includes its being a Grothendieck abelian category), a Gabriel-Popescu-type theorem is proved. An important example is when \(\mathscr{V}\) is the category of chain complexes so that Grothendieck DG-categories are characterized. Preservation of Grothendieckness is proved under change of base by a monoidal right adjoint functor.