for every $Y = U \setminus V$, $\phi$ induces a $*$-homomorphism ${  \phi_{Y}  \colon  \mathfrak{A} ( Y )  \rightarrow  \mathfrak{B} (Y)  }$. Let ${\mathfrak{C}^{*}\text{-}\mathfrak{alg}}(X)$ be the category whose objects are $C^{*}$-algebras over $X$ and whose morphisms are $X$-equivariant $*$-homomorphisms.

For any subspace $Y$ of $X$, we get a canonical covariant functor $i_Y \colon {\mathfrak{C}^{*}\text{-}\mathfrak{alg}}(Y) \to {\mathfrak{C}^{*}\text{-}\mathfrak{alg}}(X)$ given by $i_Y(\mathfrak A) (U) = \mathfrak A(Y\cap U)$, for every $U\in \mathbb O(X)$. In particular, if $Y=\{ x\}$ for a point $x\in X$ we obtain the functor ${  i_x  \colon  {\mathfrak{C}^{*}\text{-}\mathfrak{alg}}\cong {\mathfrak{C}^{*}\text{-}\mathfrak{alg}}(\{x\}) \rightarrow  {\mathfrak{C}^{*}\text{-}\mathfrak{alg}}(X)  }$.

Suppose that $X$ is a sober space. Let $\mathfrak A_1 \xrightarrow{\lambda_{1,2}} \mathfrak A_2 \xrightarrow{\lambda_{2,3}} \dots$ be an inductive system with each $\mathfrak A_n$ a $C^\ast$-algebra over $X$ and $\lambda_{k,k+1}$ an $X$-equivariant $\ast$-homomorphism. We will say that $(\mathfrak A_k, \lambda_{k,k+1})$ is an inductive system of $C^\ast$-algebras over $X$. By exactness of the $C^\ast$-algebra inductive limit functor, the inductive limit $\mathfrak A$ is canonically a $C^\ast$-algebra over $X$ for which the induced $\ast$-homomorphisms $\iota_k \colon \mathfrak A_k \to \mathfrak A$ are $X$-equivariant.

\[l:Xlimit\] Let $(\mathfrak A_k,\lambda_{k,k+1})$ and $(\mathfrak B_k,\mu_{k,k+1})$ be inductive systems of $C^\ast$-algebras over $X$ and let $\mathfrak A$ and $\mathfrak B$ be the respective inductive limits. Suppose that 