An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the high order relations of how objects fit together, but I can't see how the $Q$-construction is the natural medium for this. 


$\bullet$ $0 \to B \to A \to C \to 0$ means that $A$ is put together in some way from $B$ and $C$. Letting $E$ denote the category of short exact sequences in $P$ (with obvious morphisms), we see that this holds in the large as well: If $p$ (resp. $q$) is the projection functor $E \to P$ that sends a s.e.s. to the first (resp. last) object in the sequence, then $(K\_i(p), K\_i(q)): K\_i(E) \to K\_i(P) \oplus K\_i(P)$ is an isomorphism.


$\bullet$ Moreover, results like devissage, localisation, and resolution are also easily seen to be reassurances that we are distilling a very sensible notion of put-together-edness.



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> Can anyone offer a (non-circular) reason as to why the Q-construction is *right*?
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I don't see the sense or utility considering a category in which a morphism from $X$ to $Y$ is an isomorphism of $X$ with an (admissable) subquotient of $Y$.



What is the Q-construction, metaphysically?
There's an interesting motivation in the paper by G. Segal: 


*K-homology theory and algebraic K-theory.* K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), pp. 113–127. Lecture Notes in Math., Vol. 575, Springer, Berlin, 1977


It is well-known that the space of **Fredholm operators** gives a model for
**topological** K-theory $\Bbb Z \times BU$. 


According to Segal, if $C$ is an exact category, a morphism of the Q-construction $QC$ is akin to a Fredholm operator.


So algebraic K-theory via the Q-construction mimics topological K-theory defined via Fredholm operators.


