Proof that the determinant of the Hermitian conjugate of $A$ is the complex conjugate of the determinant of $A$

Question

While revising for an exam, I came across the following proof that the determinant of a unitary matrix $A$ is $\pm 1$:

$$1=\det(I)=\det(A^{\dagger}A) = \det(A)^{*}\det(A)=|\det(A)|^2 $$

This seems to imply that $\det(A^{\dagger})=(\det(A))^*$, but I can't seem to find a proof for this anywhere. Why is this relation the case?