I am trying to understand direct limits in the category of $C^\*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts: 



> 
> Let $(A\_n,f\_n)$ be a direct sequence of $C^\*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim\_{\rightarrow} M\_2(A\_n)=M\_2(\lim\_{\rightarrow} A\_n)$$
>  Where for the system $M\_2(A\_n)$ the connecting maps are the natural maps obtained using $f\_n$ componentwise.
> 
> 
> 


I do feel like the result should be true, but I don’t really have an argument. Any ideas?



Behaviour of direct limit with matrices
You’re asking whether the functor $M\_2$ on Banach spaces preserves colimits of direct sequences.


(In case you’re not familiar with the categorical terminology I’m using here, don’t be put off — it’s not very deep, it’s just useful packaging-up of the kind of properties you’re discussing in the question, and hopefully this answer will give an idea of why this sort of categorical language is helpful.)


The forgetful functor $U$ from C\*-algebras to Banach spaces *preserves and reflects limits*, so it’s enough to check the isomorphism $\varinjlim M\_2(A\_n) \cong M\_2(\varinjlim A\_n)$ on the underlying Banach spaces.


But at the level of Banach spaces, $U(M\_2(A)) \cong U(A) \oplus U(A) \oplus U(A) \oplus U(A)$ — i.e. the *coproduct* of 4 copies of $U(A)$. So it’s enough to show that the functor $X \mapsto X \oplus X \oplus X \oplus X$ on Banach spaces preserves colimits of direct sequences.


But this is automatic, because *colimits commute with colimits*: concretely, check the universal properties a sequential colimit of coproducts, versus a coproduct of sequential colimits, and notice that they agree.


