Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category L, representing the signature in some sense, is given, together with a set \Sigma of morphisms having finitely presentable domains and codomains. And our category K is the full subcategory of L on all objects K orthogonal to ) Supported by the Grant Agency of the Czech Republic under the grant No. 201/99/0310. 1 each s : X ! X 0 in \Sigma (notation: K ? s), which means that every morphism f : X ! K uniquely factors through s; notation: K = \Sigma ? . Such subcategories K of L are called in [AR] the !-orthogonality c