We answer the following two questions: (I) What order complete Banach lattices E have the property: if K is a compact Hausdorff space, then every weakly compact operator C(K)\(\to E\) is regular (in the sense of Riesz space theory)? (II) What weakly Fatou order complete Banach lattices E have the property: if K is a compact Hausdorff space and if T is a regular compact operator C(K)\(\to E\), then \(| T|\) is weakly compact?