Let's start this topic with a metaphor. If you are not familiar with the game of billiards, take a look here to get a basic idea. https://en.wikipedia.org/wiki/Cue_sports

Sometimes in life, there are many details layered on top of something that make it seem as if things are different when they are really the same. Let's take the game of 8-ball. In this game, we have an underlying game of physics: hit balls in the holes. However, we have layered rules on top which make it seem as if there are differences in a billiard ball painted with stripes, and one painted a solid color. When we remove the rules of 8-ball (the extra details), we see that billiard balls are very much the same regardless of how they are painted. That is to say, if our only goal is to hit balls in holes (no extra details), all the balls are the same for these intents and purposes. Once we have removed all the extra details, the billiard balls are all isomorphic.

Isomorphism of objects in category theory is about "sameness". In the previous article we saw that categories are simply property-less objects with arrows between them. Two objects in a category are isomorphic if they are the "same" for all intents and purposes. Strictly speaking, two objects are isomorphic if there exists in the category an arrow from object a to object b, and an arrow from object b to object a that is the inverse of the first arrow.

In the previous article we talked about the idea of "identity". To recap, the identity morphism is the morphism that starts from object a and points back to object a. In terms of isomorphism of objects, if you were to compose the two arrows mentioned in the paragraph above, you would have the identity morphism. That is to say, if an arrow goes from object a to object b and another arrow goes from object b to object a that is the inverse of the first arrow and you compose these two arrows together, you have an arrow from object a to object a, the identity morphism!

Isomorphism is important in category theory because it allows us to disregard extra details that make categories seem different and look at them as the same. This allows us to focus on the properties of categories and not the nuances of their application.

Remember, we talk a lot about objects and arrows and this is not a metaphor. This is because category theory is EXACTLY about objects and arrows. It is so abstract because these primitive objects have no inherent properties. This is also why category theory describes the mechanics of so many things. All the information is encoded in the morphisms.