Incorrect usage!
The proper term is temperate phage.
A lysogen is a bacterium that is infected with a temperate phage where the phage is displaying a lysogenic cycle. Lysogens are called lysogens because they are "lysis generating" bacterial strains. Since phages are not bacterial strains, they cannot be described as lysogenic.
Temperate phages nonetheless can display lysogenic cycles as well as lysogenic infections and give rise to lysogenic bacteria/lysogenic cultures.
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