I'm working through the exercises in "Category Theory for Computing Science" by Barr and Wells, and I'm a little confused about an early problem (though it could just come down to notation). Question 3 in section 1.2 reads:
- Let S and T be two disjoint sets, and let V be a set.
a. Let $\phi : \operatorname{Hom}(S, V ) × \operatorname{Hom}(T, V ) \rightarrow \operatorname{Hom}(S \cup T, V)$ be the mapping that takes a pair $(f : S \rightarrow V, g : T \rightarrow V )$ to the function $\langle f|g \rangle : S \cup T \rightarrow V$ defined by
$$\langle f|g \rangle(x) = \left\{ \begin{array}{lr} f(x) & : x \in S\\ g(x) & : x \in T\end{array} \right.$$
Show that φ is a bijection.
b. If you set $V = S \cup T$ in (a), what is $\phi(id_{S \cup T})$?
I can do the first part, but I'm a little confused about $\phi(id_{S \cup T})$. $id_{S \cup T}$ is a function $S \cup T \rightarrow S \cup T$, whereas $\phi$ takes a pair of functions $S \rightarrow S \cup T$ and $T \rightarrow S \cup T$, so how is $\phi(id_{S \cup T})$ meaningful?
If it helps, the answer from the answer book is "it is $(i,j)$ where $i$ is the inclusion of $S$ in $S\cup T$ and $j$ is the inclusion of $T$ in $S \cup T$". That only makes sense to me if the question is asking about $\phi^{-1}(id_{S \cup T})$ (so $\phi((i,j)) = id_{S \cup T}$), but that doesn't seem to be the case.