This is from Advanced Linear Algebra by S.Roman.
Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon A\to X \mid X\in\mathcal S \} $$ be a family of functions, all of which have domain $A$ and range a member of $\mathcal S$. Let $$ \mathcal H = \{ \tau \colon X \to Y \mid X,Y \in \mathcal S \} $$ be a family of functions with domain and range in $\mathcal S$.
We assume that $\mathcal H$ has the following structure:
- $\mathcal H$ contains the identity function $\iota_S$ for each member of $\mathcal S$.
- $\mathcal H$ is closed under composition of functions, which is an associative operation.
For any $\tau \in \mathcal H$ and $f \in \mathcal F$, the composition $\tau\circ f$ is defined and belongs to $\mathcal F$.
We refer to $\mathcal H$ as the measuring family and its members as measuring funcitons.
I do not understand property 3 in the definition.
However this actually contradicts the diagram (consider picking $\tau_2$ and $f_2$, then $\tau_2\circ f_2$ isn't defined)
My question is: what does it mean to say?
I've tried searching for terms like "measuring family universal property" but I either get property management stuff, stuff on measuring family functioning, or a link to Google books. I've looked in the index and scanned through Categories for the working mathematician and haven't found it either. I'd like to be sure about this before proceeding (and not have to guess looking at proofs and the desired properties)