Let $X$ and $Y$ be smooth manifolds equipped with fixed smooth positive densities, and let $E \to Y$ and $F \to X$ be complex vector bundles equipped with Hermitian fiber metrics. Let $C \subset T^*X \setminus 0 \times T^*Y \setminus 0$ be a homogeneous canonical relation, with the convention that a local phase function $\phi(x,y,\theta)$ parametrizes $C$ by sending $(x,y,\theta)$ with $\partial_\theta\phi(x,y,\theta)=0$ to $(x,\partial_x \phi(x,y,\theta); y,-\partial_y \phi(x,y,\theta))$. Define the transposed canonical relation $C^t \subset T^*Y \setminus 0 \times T^*X \setminus 0$ by $C^t := \{(y,\eta; x,\xi) : (x,\xi; y,\eta) \in C\}$. Fix the standard principal-symbol convention for Fourier integral operators in which the principal symbol is the leading homogeneous amplitude tensored with the phase-induced half-density and Maslov factor, modulo the usual transformation law under equivalent phase functions. If $A \in I^m(X,Y;C;\operatorname{Hom}(E,F))$ is a properly supported Fourier integral operator, then its adjoint, defined with respect to the chosen densities and Hermitian pairings, satisfies $A^* \in I^m(Y,X;C^t;\operatorname{Hom}(F,E))$. Moreover, under the identification of the half-density and Maslov symbol bundles induced by the diffeomorphism $C \to C^t$, $(x,\xi;y,\eta) \mapsto (y,\eta;x,\xi)$, the principal symbol of $A^*$ is the fiberwise Hermitian adjoint of the principal symbol of $A$. In the scalar case, this is complex conjugation.