Let $(I,\leq)$ be a directed set, and let $(A_i,\varphi_{ij})_{i\leq j}$ be a directed system of $C^*$-algebras, where each $\varphi_{ij}:A_i\to A_j$ is a $*$-homomorphism, $\varphi_{ii}=\operatorname{id}_{A_i}$, and $\varphi_{jk}\circ\varphi_{ij}=\varphi_{ik}$ whenever $i\leq j\leq k$.

Then there exist a $C^*$-algebra $A$ and $*$-homomorphisms $\varphi_i:A_i\to A$ such that $\varphi_j\circ\varphi_{ij}=\varphi_i$ for all $i\leq j$, and such that $(A,(\varphi_i)_{i\in I})$ satisfies the following universal property: for every $C^*$-algebra $B$ and every compatible family of $*$-homomorphisms $\psi_i:A_i\to B$, meaning $\psi_j\circ\varphi_{ij}=\psi_i$ for all $i\leq j$, there exists a unique $*$-homomorphism $\psi:A\to B$ such that $\psi\circ\varphi_i=\psi_i$ for every $i\in I$.

Moreover, if every connecting map $\varphi_{ij}$ is injective, then every canonical map $\varphi_i$ is injective. In that case, after identifying each $A_i$ with its image in the limit, $A$ is the norm closure of the increasing union $\bigcup_{i\in I}\varphi_i(A_i)$.