Injective cogenerators among operator bimodules

Given C${}^{\ast }$-algebras $A$ and $B$ acting cyclically on Hilbert spaces $\text{h}$ and $\text{k}$, respectively, we characterize completely isometric $A,B$-bimodule maps from $\text{bkh}$ 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${}^{\ast }$-subalgebra of $\text{al}(X)$ (adjointable left multipliers on $X$), is equal to its relative double commutant in $\text{al}(X)$, then $A$ must be a W${}^{\ast }$-algebra.