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In Topology by Munkres; one problem is stated as proving the following theorem:

Let $J$ be a well ordered set, let $C$ be a set. Let $F$ me the set of all functions mapping sections of $J$ into $C$. Given a function $\rho : F \to C$ , there exists a unique function $h: J \to C$ such that $h(\alpha) = \rho(h | S_\alpha)$ for each $\alpha \in J$

I'm trying to understand what $\rho(h | S_\alpha)$ means. Is $h|S_\alpha$ supposed to be interpreted as the function $h$ acting on the limited domain of the section $S_\alpha$ (thus making it a function mapping a section of $J$ into $C$)? Usually I read the vertical line differently, for instance I'd read $\{x | x > 0 \}$ as $x$ where $x > 0$.

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    Yes, it looks like the restriction of $h$ to $S_{\alpha}$, often written using a subscript like this: $h|_{S_{\alpha}}$ . If $h:J\to C$, then $h|_{S_{\alpha}}: S_{\alpha}\to C$ via $h|_{S_{\alpha}}(x) = h(\iota(x))$ where $\iota:S_{\alpha}\hookrightarrow J$ is inclusion. – 2017-01-18
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    Some recursive definition uses all of its "previous" evaluations (e.g. $a_{n+1} := a_1+\cdots + a_n$.) This is a reason why we use $h|S_\alpha$ in the recursive definition; $h|S_\alpha$ as a set contains all of $h(\beta)$ for all $\beta<\alpha$. – 2017-01-19

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