On a problem of Gabriel and Ulmer

We present a locally finitely presentable category with a finitely presentable regular generator image and a finitely presentable object A, such that A is not a coequalizer of morphisms whose domains and codomains are finite coproducts of objects in image, thereby settling a problem by Gabriel and Ulmer. We also show that in λ-orthogonality classes in image (category of image-sorted τ-algebras) for a λ-ary signature τ, λ-presentable objects have a presentation by less than λ generators and relations and use this to exhibit an example of a reflective subcategory of a locally finitely presentable category which is closed under directed colimits, but not a aleph, Hebrew0-orthogonality class, disproving a characterization of λ-orthogonality classes in the book by Adámek and Rosický.