Let $L_{p_i,\phi_i}$ $(i=1,2,3)$ be Morrey spaces.
A function $g$ is called a pointwise multiplier
from $L_{p_1,\phi_1}$ to $L_{p_2,\phi_2}$,
if the pointwise product $fg$ belongs to $L_{p_2,\phi_2}$
for each $f\in L_{p_1,\phi_1}$.
We denote by $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})$ the set of all pointwise multipliers from $L_{p_1,\phi_1}$ to $L_{p_2,\phi_2}$.
A sufficient condition on $p_i$ and $\phi_i$ ($i=1,2,3$)
for $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})=L_{p_3,\phi_3}$
was given in \cite{Nakai1997b}.
In this paper, we give a necessary condition.
In connection with these conditions,
we also give sufficient conditions
for $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})=\{0\}$.