Why are these linear matrix inequalities equivalent?

Question

Given a square matrix $A$, we want to find the symmetric matrix $P$ such that $$ A^TP + PA < 0, P > 0 \tag{1} $$ In a paper that I'm reading, the authors write that the strict linear matrix inequalities in $(1)$ can be replaced by the following non-strict linear matrix inequalities because "the inequalities in $(1)$ are homogeneous in $P$": $$ A^TP + PA \leq -I, P \geq I \tag{2} $$ If $P$ satisfies the non-strict inequalities in $(2)$, does it necessarily satisfy the strict inequalities in $(1)$ (and vice versa)? If so, why?