Hovey book Model Categories

Question

In his book Model Categories Hovey uses often the symbol $I$ as in the snippet below. But on the page 14 he says that $I$ is a set and on the page $30$ that $I$ is a set of class of maps in $\cal C.$ What is the meaning of $I$ in the snippet below and then what does it mean $B\times I$ and $(i_0,i_1)$ in $B\coprod B\to B\times I$ in the diagram there ?

Answer 1

$B \times I$ does not literally mean the product of $B$ and $I$ here, unfortunately. ${} \times I$ is to be read as an atomic symbol and denotes a certain functor (obtained from the functorial factorisation of $B \amalg B \to B$) applied on the right. The morphisms $i_0, i_1 : B \to B \times I$ are obtained from the same factorisation by precomposing with the two coproduct insertions $B \to B \amalg B$.

Incidentally, $B^I$ does not literally mean the exponential of $B$ by $I$ here.