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I have split off an entry homotopy in a model category from homotopy and then spelled out statement and proof of the basic lemmas.
In the lemma
Let $X$ be cofibrant. If there is a left homotopy $f \Rightarrow_L g$ then there is also a right homotopy $f \Rightarrow_R g$ (def. \ref{LeftAndRightHomotopyInAModelCategory}) with respect to any chosen path object.
I believe the chosen path object must also be good. (Otherwise, $Y$ itself is a non-good path object, and a right homotopy with respect to it would say $f=g$.)
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