Formally smooth morphism

Okay kiddo, let's talk about formally smooth morphisms. Imagine that you have two things, let's call them A and B, and there's a special way to turn A into B. We call this a "morphism" between A and B.

Now, let's say that you have another thing called C, and you wonder if there's a way to turn it into another thing called D using the morphism between A and B. But there's a catch - you can only change things a little bit at a time (this is called an "infinitesimal change").

In that case, if there's a way to make these little changes without messing things up too much, we say that the morphism between A and B is "formally smooth" in relation to C and D. It's like being able to slide one thing into another, really gently and smoothly, so that it fits perfectly.

This idea is important in math when studying how different things are related to each other, and how they can change in specific ways. It can help us understand things like how shapes can transform into each other, or how one type of equation can be changed into another.