# Why does Spanier give two seemingly different definitions of the Category of pairs?

In his book Algebraic Topology (p.16), Spanier writes:

So from this, I understand that its objects are the injective morphisms of $\mathscr{C}$ and its morphisms with domain $i_{1}:A_{1}\rightarrow B_{1}$ and range $i_{2}:A_{2}\rightarrow B_{2}$ are pairs of morphisms of $\mathscr{C}$ , $(g:A_{1}\rightarrow A_{2},h:B_{1}\rightarrow B_{2})$ , such that $h\circ i_{1}=i_{2}\circ g$ . Composition is define componentwise. So far, I understand everything.

But this is when I get confused. On the next page, he writes:

So now he talks about pairs of the objects of $\mathscr{C}$ object consisting of a set, a subset of it, and the inclusion mapping. He says that this is an object of the category of pairs and he says that a morphism $f:(X,A)\rightarrow (Y,B)$ whose image is mapped into $B$ is a morphism in the category of pairs. Here are my questions:

1. Are these two definitions different?

2. In the second definition, he only considers inclusion mappings. While in the first definition, he talks about injections in general. Why does he do this?

3. Could it be that Spanier is definining Categories of pairs in a horrible way? If so, could someone provide me with a clear definition?