Summary: Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls to stochastic “outputs”. A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations is given in this context. A notion of compatibility between inputs and outputs is critical in relating the general result to its classical forebears. The usual formulation of stochastic differential equations driven by semimartingales does not require compatibility, so a notion of partial compatibility is introduced which does hold. Since compatibility implies partial compatibility, classical strong uniqueness results imply strong uniqueness for compatible solutions. Weak existence arguments typically give existence of compatible solutions (not just partially compatible solutions), and as in the original Yamada-Watanabe theorem, existence of strong solutions follows.