# Constructing an injective resolution for a bounded below cochain complex

Let $$\mathcal{A}$$ be an abelian category with enough injectives. If $$X^{\bullet}$$ is a bounded below complex, it is a well known fact that you can obtain a bounded below complex $$I^{\bullet}$$ of injective objects in $$\mathcal{A}$$ and a quasi-isomorphism $$\epsilon: X^{\bullet} \rightarrow I^{\bullet}$$. The answer to the question here gives a particularly nice construction of this, but I am not quite sure I follow the logic.

The construction proceeds inductively. It shows that if you have the complex $$X^{\bullet}$$, and some complex of injectives $$I^{\bullet}$$ with morphisms $$\{ f^{n}: X^{n} \rightarrow I^{n} \text{ for all } n \leq a \}$$, then you can inductively produce $$f^{n+1}$$ so that it is a quasi-isomorphism in each degree.

This is all well and good, but I don't see how you pass from the finite to the infinite. This allows you to construct a sequence of truncated above chain complexes, with quasi-isomorphisms in higher and higher degrees, but unless you have some notion of "taking the limit" then I don't see how this produces a morphism of complexes. The obvious notion here would be some kind of direct limit, but this construction is supposed to work even when direct limits may not be exact.