If I have a project licensed under the MIT License, and I find a snippet of code I want to integrate into my project, also licensed under MIT, how do I fulfill the conditions of the project the snippet originated from?

My project has a license file in the root of the source code, as required by the MIT License.

From choosealicense.com under "How to apply this license":

Create a text file (typically named LICENSE or LICENSE.txt) in the root of your source code and copy the text of the license into the file. Replace [year] with the current year and [fullname] with the name (or names) of the copyright holders.

If I want to incorporate someone's MIT licensed snippet, do I need to replace the license file? Do I need to add a license file?