Pullback (category theory)

Okay, so imagine that you have a puzzle with three pieces - a square, a triangle, and a circle. But the circle is missing a small piece in the middle.

Now let's say that you have another puzzle with just two pieces - a circle and a triangle. And the circle piece fits perfectly into the missing piece in the first puzzle.

When we talk about a "pullback" in category theory, we are talking about a similar situation where we have two puzzles (categories) and we find pieces (objects) that fit together perfectly.

More specifically, a pullback is a way to connect two objects in different categories so that they fit together in a certain way. For example, let's say we have two categories - one for animals and one for sounds. We have an object in the animal category (let's call it "dog") and an object in the sound category (let's call it "bark"). A pullback in this situation would be finding the connections between the "dog" object and the "bark" object so that they fit together in a specific way, maybe by saying that "dog" is the object that makes the "bark" sound.

So, a pullback is like finding the missing piece in a puzzle and connecting two things together in a very specific and precise way.