The space \({\mathcal K}(E,F)\) of compact operators is, in general, not complemented in the space \({\mathcal L}(E,F)\) of all continuous operators (\(E\), \(F\) Banach spaces). The author shows, however, that the dual \({\mathcal K}(E,F)'\) is isomorphic to a complemented subspace of \({\mathcal L}(E,F)'\) (the complement being the annihilator of \({\mathcal K}(E,F)\)) – provided that every operator \(T:E\to F\) factors through a Banach space \(Z_T\) with the bounded approximation property and separable dual. For \(F'\) being separable this generalizes a former result of J. Johnson \(Z_T= F\) for all \(T\) in [J. Funct. Anal. 32, 304-311 (1979; Zbl 0412.47024)] and for \(E=P\) and \(F=P'\) (where \(P\) is any separable Pisier space hence \(Z_T=\ell_2\)) this is also due to K. John [Note Mat. 12, 69-75 (1992; Zbl 0802.46015)].