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 below, 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/)