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We study when the heart of a t-structure in a triangulated category with coproducts is AB5 or a Grothendieck category. If satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t …