A Banach space \(X\) is said to have the approximation property if its identity operator can be uniformly approximated on compact subsets of \(X\) by finite-rank operators. If the identity is allowed to be approximated by compact operators (instead of finite-rank operators), then \(X\) is said to have the compact approximation property (C.A.P. for short). According to [\textit{A. Lima, O. Nygaard} and \textit{E. Oja}, Isr. J. Math. 119, 325--348 (2000; Zbl 0983.46024)], \(X\) has the approximation property if and only if the finite-rank operators \(F(Y,X)\) form an ideal in the Banach space \(W(Y,X)\) of weakly compact operators from \(Y\) to \(X\), for all Banach spaces \(Y\). It was furthermore proved in [op.~cit.] that the C.A.P. of \(X\) implies that the compact operators \(K(Y,X)\) form an ideal in \(W(Y,X)\) for all Banach spaces \(Y\), but the converse does not hold. In the present article, the method of [op.~cit.] is developed further to prove that \(X\) has the C.A.P. if and only if for all Banach spaces \(Y\) and for all \(T\in W(Y,X)\), the operators \(ST\) with \(S\in K(X,X)\) form an ideal in the subspace generated by them and \(T\). Relying on this theorem and results from [op.~cit.], the authors characterize the C.A.P. of \(X\) in terms of the approximability of the identity operator in the strong operator topology by compact operators \(S\) on \(X\) such that \(\| ST\| \leq\| T\| \) for \(T\in W(Y,X)\). Similar results for the dual space \(X^\ast\) are obtained. Relying also on [\textit{A. Lima, E. Oja}, J. Aust. Math. Soc. 77, No. 1, 91--110 (2004; Zbl 1082.46016)] and [\textit{V. Lima}, Math. Scand. 93, No. 2, 297--319 (2003; Zbl 1067.46027)], the C.A.P. of \(X^\ast\) given by conjugate operators is compared with the C.A.P. of \(X^\ast\) in terms of ideals and equivalent renormings of \(X\).