On Banach ideals satisfying $c_0(\mathcal{A}(X,Y))=\mathcal{A}(X,c_0(Y))$

We characterize Banach ideals $[\mathcal{A},a]$ satisfying the equality $c_0(\mathcal{A}(X,Y))= \mathcal{A}(X,c_0(Y))$ for all Banach spaces $X$ and $Y$. Among other results we have proved that $\mathcal{K}$ (the normed operator ideal of all compact operators with the operator norm) is the only injective Banach ideal satisfying the equality