Garrett Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944) 764–768 pdf

For another version see

Walter Tholen, Injective objects and cogenerating sets, J. Algebra 73(1) (1981) 139–155 doi

Walter Tholen, Birkhoff’s theorem for categories, In: Banaschewski, B. (eds) Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics 915, Springer 1982, doi

(surveyed as Th. 3.5 in) M. Mehdi Ebrahimi, Mahdieh Haddadi, Mojgan Mahmoudi, Injectivity in a category: an overview on smallness conditions, Categories and General Algebraic Structures with Applications 2:1 (2014) 83–112 pdf