Injective cogenerators among operator bimodules

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Given C*-algebras A and B acting cyclically on Hilbert spaces and, respectively, we characterize completely isometric A,B-bimodule maps from into operator A,B-bimodules. We determine cogenerators in some classes of operator bimodules. For an injective cogenerator X in a suitable category of operator A,B-bimodules we show: if A, regarded as a C*-subalgebra of (X) (adjointable left multipliers on X), is equal to its relative double commutant in (X), then A must be a W*-algebra.