Summary: We study (possibly non-self-adjoint) subspaces of \(\mathcal{L}(H)\) together with the induced partially defined involutions and the sequence \(\{M_n(X)\}_{n\in\mathbb{N}}\) of ordered normed spaces. These are called “non-self-adjoint OSs”. Two particular concerns are the abstract characterization of non-self-adjoint OSs and the injectivity in the category of non-self-adjoint OSs (known as “MOS-injectivity”). In order to define MOS-injectivity, we need the notion of “unitalization” and “MOS-subspace”. We show that in the case of an operator algebra \(A\) (which is a non-self-adjoint OS), its unitalization coincides with another unitalization defined in [D.P. Blecher and C. Le Merdy, Operator algebras and their modules – an operator space approach, London Mathematical Society Monographs. New Series 30; Oxford Science Publications. Oxford: Oxford University Press (2004; Zbl 1061.47002)] if and only if a form of Stinespring dilation theorem holds for \(A\). On the other hand, through the study of injective objects, we define “MOS-injective envelopes” and “MOS-\(C^{*}\)-envelopes” of non-self-adjoint OSs. It is interesting to note that in the case of a unital operator space \(V\) (which is a non-self-adjoint OS), its “MOS-injective envelope” does not need to coincide with the ordinary injective envelope \(V_{\text{inj}}\) (they are the same if \(V\) is an operator system), but one can identify \(V_{\text{inj}}\) with the MOS-injective envelope of a unital operator system associated with \(V\). Similarly, the “MOS-\(C^{*}\)-envelope” of an operator algebra \(A\) does not need to be the same as the ordinary \(C^{*}\)-envelope \(C^{*}(A)\), but one can recover \(C^{*}(A)\) as the MOS-\(C^{*}\)-envelope of a unital operator system associated with \(A\).