A universal characterisation of the Bochner integral

We provide a universal characterisation of the Banach spaces of equivalence classes of $L^p$ Bochner integrable functions, under equality almost everywhere, as initial objects in appropriate categories of bifunctors, for all $p\in [1,\mathrm{\infty })$. As a by-product of our construction, we also obtain a unique characterisation of the Bochner integral.