  We establish classical and categorical Howe dualities between the Lie
superalgebras $\mathfrak{p}(m)$ and $\mathfrak{p}(n)$, for $m,n \geq 1$. We
also describe a presentation via generators and relations as well as a Kostant
$\mathbb{Z}$-form for the universal enveloping superalgebra
$U(\mathfrak{p}(m))$.
