Quasibarrelled, barrelled and bornological tensor products of locally convex spaces are studied. A device - called desintegration lemma - is developed for the most difficult case, that of the injective topology. The main theorems are (proper) extensions of results about barrelledness properties of spaces of vector-valued continuous functions due to [\textit{J. Mendoza Casas}: Simon Stevin 57, 103-123 (1983; Zbl 0517.46029)] in terms of topological tensor products.