In his book, Category Theory, Awodey writes the following about it:
Next, we are going to see the categorical definition of a product of two objects in a category. This was first given by Mac Lane in 1950, and it is probably the earliest example of category theory being used to define a fundamental mathematical notion.
In the last editions of Algebra and Categories for the Working Mathematician, the notion of product in a category is presented (of course). I was wondering how did MacLane first arrive at and where did he publish this notion?
 Awodey, Steve: Category Theory. Oxford University Press, 2010.
 Mac Lane, Saunders and Birkhoff, Garrett: Algebra. AMS Chealsea Publishing, 1999.
 Mac Lane, Saunders: Categories for the Working Mathematician. Springer, 1998.