On some isomorphism on the category of $b$-spaces

Given a nuclear $b$-space $N$, we show that if $\Omega$ is a finite or $\sigma$-finite measure space and $1\leqslant p\leqslant\infty$, then the functors $L_{\textup{loc}}^p(\Omega,N\varepsilon\cdot)$ and $N\varepsilon L^p(\Omega,\cdot)$ are isomorphic on the category of $b$-spaces of L. Waelbroeck.