In the book More concise algebraic topology on the page 213 they write
We think of fibrations as analogous to epimorphisms.
BUT Hovey on the page 51 says $f$ is a fibration if it is in $J-inj$. My question is simple: how epimorphisms (rather than monomorphisms) are all of $J-inj$ ? I.e. how can $J-inj$ be thought of as epis?
If you read definition 2.1.7 of Hovey you see that $J$-inj is the maps with the RLP with respect to the maps in $J$. These should remind with $J$ being the inclusions $D^n\to D^n\times I$. One example of such maps are covering maps, these we would definitely want to be surjective.