The answer is positive. I found a published account with details to be chapter 6 and 7 of Handbook of Categorical Algebra 1 by Francis Borceux. Thanks to the comments, useful links that summarize how it goes are + https://mathoverflow.net/questions/85768/morita-equivalence-for-categories + https://mathoverflow.net/questions/163638/morita-equivalence-via-kan-extension + [Qiaochu Yuan's wonderful writeup](https://qchu.wordpress.com/2015/05/07/tiny-objects/)