Probably this is rather trivial, still I would like to have a good insight on it :
Let $\mathcal{F}$ be an ultrafilter on $\mathbb{N}$ and consider an indexed family of sets $\lbrace X_{n}\rbrace_{n\in \mathbb{N}}$ of sets. Considering $\mathcal{F}$ as a partially ordered category, one can define a functor $\Phi :\mathcal{F}^{op}\rightarrow$ Set, setting $\Phi(A)=\prod_{n\in A}X_n$ and for any given map $f:B\rightarrow A$ in $\mathcal{F}^{op}$, $\Phi(f)$ is the projection on indexes. Then, one can check that the ultraproduct $\prod_{\mathcal{F}}X_n$ is actually the colimit of $\Phi$.
My question is : If instead of Set we consider the category of C*-algebras, can I understand the usual definition of ultraproduct of C*-algebras in a similar way ? If so, I would appreciate some details or references in the answer.
NOTE : The usual definition of ultraproduct of a family $\lbrace\mathcal{A}_n\rbrace_{n\in\mathbb{N}}$ of C*-algebras, along an ultrafilter $\mathcal{F}$ on $\mathbb{N}$ is as the quotient $l^{\infty}/c_0^{\mathcal{F}}$, where $c_0^{\mathcal{F}}=\lbrace (a_n)\in l^{\infty} : \lim_{\mathcal{F}}||a_n||=0\rbrace$.