  We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is
octahedral for every non-zero Banach space $Y$. They are those satisfying that,
for every finite dimensional subspace $Z$, $\ell_\infty$ can be
finitely-representable in a part of $X$ kind of $\ell_1$-orthogonal to $Z$. We
also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if,
$L(\ell_p^n,X)$ is octahedral for every $n\in\mathbb N$ and $1<p<\infty$.
Finally, we find examples of Banach spaces satisfying the above conditions like
$\Lip(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet
property.
