Inverse limits are projective limits

Question

I'm learning about projective limits, and one definiton (which you will find on Wikipedia) is done with a projective system over groups, call it $(X_i, \varphi_{i,j})$. And they define the inverse limit as: $$\varprojlim Y_i = \\{ y\in \prod\limits_{i\in I} Y_i\mid y_i = \varphi_{ij} (y_j) \text{ for all } i\leq j\in I \\}$$ My question now, or more accurately my request, is if anyone here knows of a resource, be it a textbook or otherwise, that shows that the above definition is in fact a projective limit, in the universal property sense of the term. Just a point in the right direction would be super appreciated! Thanks!