Coequalizer

Have you ever seen a bunch of toys scattered all over your bedroom floor? It can be quite messy and confusing to find what you need. Imagine you have two big baskets, one with some cars and the other with some dolls in them. When you pick up a toy and decide where it goes, you can either put it with the other cars or with the other dolls based on what type it is.

Coequalizer is a way to do something similar with mathematical objects called subsets. Let's say there are two sets, A and B. Each set can have many subsets, like the basket of cars and basket of dolls. Sometimes there are functions, which take elements from set A to set B in a very special way. These functions are called morphisms.

Now, imagine you have two different morphisms that match up the same elements in set A and B. This is like having two identical toy cars, one in each basket. They are the same car, just in different places.

A coequalizer is like taking those matching morphisms and putting all the elements that match into a new set. It's like cleaning up your toys and putting them away in the right spot. The coequalizer makes sure everything that is the same is together and not scattered.

Overall, a coequalizer is a way to organize mathematical objects in a set, just like baskets help organize toys.