I have a question about a path object in the context of model categories. For what Hovey says see the first snippet below. But here on page 9 the definition is different: Henry says (see the second snippet below) that the composite of $P_X Y \twoheadrightarrow Y \times_X Y \twoheadrightarrow Y$ is an acyclic fibration, but Hovey says $Y \twoheadrightarrow P_X Y$ is a weak equivalence.
What is the relationship between these two definitions?
First snippet (Mark Hovey):
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Second snippet (Simon Henry):
First, note that they apply in different contexts, in Hovey we make no assumptions about whether or not $Y$ is fibrant, and in Henry, we have a relative notion defined for a fibration $Y\newcommand\fib\twoheadrightarrow \fib X$.
Also a note, since it's not explicit in Henry, $Y\to P_XY$ is a weak equivalence by 2-of-3, since $Y\to P_XY\fib Y\times_X Y \to Y$ composes to the identity map, so it's equivalent to assume either $Y\to P_XY$ or $P_XY\fib Y\times_X Y\fib Y$ is a weak equivalence.
Now in the common context, $X=*$, $Y$ fibrant, we can ask whether the two definitions agree, and in fact they do, by the observation in the previous paragraph.