  We derive formalism for determining $\textbf{2} + \mathcal J \to \textbf{2}$
infinite-volume transition amplitudes from finite-volume matrix elements.
Specifically, we present a relativistic, model-independent relation between
finite-volume matrix elements of external currents and the physically
observable infinite-volume matrix elements involving two-particle asymptotic
states. The result presented holds for states composed of two scalar bosons.
These can be identical or non-identical and, in the latter case, can be either
degenerate or non-degenerate. We further accommodate any number of
strongly-coupled two-scalar channels. This formalism will, for example, allow
future lattice QCD calculations of the $\rho$-meson form factor, in which the
unstable nature of the $\rho$ is rigorously accommodated.
