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I’m looking for advice on how to call classes of morphisms $M$ whose orthogonality and injectivity classes coincide: $M$ has this property if every $M$-injective object $X$ is also $M$-orthogonal. Or has perhaps somebody already given a name to this property somewhere?
The concept is relevant because
a) the small object argument for sets of morphisms with this property computes the reflection into the orthogonal subcategory, and
b) every class of morphisms can be extended to a class with this property and the same orthogonality class (add $B \amalg_A B \rightarrow B$ for each map $A \rightarrow B$ in the original set).
Thanks!
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