If \(\theta\) is a compact endomorphism on a commutative, semisimple, unital Banach algebra A, and \(\theta^*\) is the adjoint of \(\theta\), it is shown by \textit{H. Kamowitz} [Pac. J. Math. 89, 313-325 (1980; Zbl 0465.46047)] that the intersection \(\cap \theta^{*n}(A')\) is finite, where A' is the set of all multiplicative linear functionals on A. In the present paper, the author characterizes compact homomorphisms \(\theta\) in a more general setting where it is defined from a \(C^*\)-algebra A into a Banach algebra B that has a continuous imbedding in B(H), the Banach algebra of bounded linear operators on some Hilbert space H. It is shown that \(\theta\) is a finite rank operator and the range of \(\theta\) is spanned by a finite number of idempotents. If, moreover, B is commutative, then \(\theta\) has the form \(\theta (x)=f_ 1(x)E_ 1+...+f_ k(x)E_ k\), where \(E_ 1,...,E_ k\) are fixed mutually orthogonal idempotents in B and \(f_ 1,...f_ k\) are fixed multiplicative linear functionals on A.