Intuition for comma category and slice category? As usual, I am not happy with the definitions until I dont gain some intuition. Could you help me and maybe provide some examples of the concepts? Thank you! My questions are:
1 Comma category = "Morphism category". Category where objects are morphisms. And morphisms are.. Morphisms between those morphisms, but in a different category (say C).
1.1 Why we need comma categories?
1.2 How to explain comma category in layman terms?
1.3 Is comma category a special type of 2-category? Or an object in a 2-category?
2 Slice category = Special case of comma category.
2.1 Why we need comma categories?
2.2 How to explain comma category in layman terms?
2.3 Simplest examples?
3 Limits and colimits - what is the intuition?
 A: Addressing all of these topics adequately is about 2 chapters in a good textbook. So, I'll concentrate on the idea for these constructions and some example that I hope will be illuminating.
The "morphism category", also known as the arrow category. In a category, you can think of the objects as meaningless entities whose sole purpose in life is to serve as the domain and codomain of morphisms. Whatever interest, or structure, we perceive the objects to have is a by-product of the morphisms they carry, and the composition of such morphisms. In a sense, the morphisms give 'life' to the objects. Once the morphisms are there, and we have a category, we can (and should) turn our attention to the morphisms themselves. The arrow category does just that. The original category embeds nicely inside the arrow category by viewing each object as the identity morphism on it. Now, the more general comma category allows us to mix this idea between different categories. It says that given some functors between categories, we can similarly 'promote' the objects into morphisms and 'enlarge' the category. It's a way to understand the functors involved.
Why we need comma categories? Because it's a simple way to build new categories out of old ones (and some functors between them).
How to explain them in layman terms? Depends on how layman you wish to be, but the above may help.
Is it a special type of 2-category, or something of that sort. Yes, and probably in more ways than one. There is a notion of comma object in a 2-category, and it captures the usual notion as a special case.
Now for slice categories. Yes, they are a special case of comma categories, but I'm going to suggest that forgetting that for a second is a good idea. An important example of comma category is as important as it is elementary: the slice category $Set/S$ of sets over a given set $S$. An object $X\to S$, just a function, can be thought of as a set $X$ of objects, each of which is assigned a colour from $S$. So, we fix the set $S$ of colours and look at all sets of $S$-coloured elements. If $S$ is a terminal object, then $Set/S\cong Set$. Rather boring. But, if $S$ is a doubleton, say $S=\{T,F\}$, the boolean truth values, then $Set/S$ is very interesting. An object is a set $X\to S$, which we may think of as a characteristic function. Thus, the objects are essentially pairs $(Y,X)$ of a set $X$ and a subset $Y$. Now, clearly there is a forgetful functor sending $X\to S$ to $X$. Now, consider an arbitrary function $f\colon X\to X'$. It is common to consider the direct image function and the inverse image function associated with $f$. The inverse image function thus takes an object $(X',Y')$ to some object $(X,Y)$. But, then this can be phrased also in the language of the comma category $Set/S$. It's thus a process that carries $X'\to S$ to some $X\to S$. The way this is done is precisely by computing the pullback. So, inverse image = pullback. Similarly, direct image = composition. This shows that very elementary operations are captured well within a suitable comma category. Much more is true; this is just the beginning of categorical logic.
So, why we need comma categories? They encode much of classical logic and extend it much beyond the classical realm. This is only one reason. How to explain them in layman terms? Again, perhaps the above helps, as well as some further examples below. Simplest examples? How about the slice category $*/Set$ of a singleton set over $Set$? It is essentially the category of pointed sets, so again we see that it is a general principle for constructing new categories from old.
Limits and colimits: the intuition is that a diagram in a category 'sees' the rest of the category in some way. The way the diagram as a whole looks at the rest of the world is by constructing a cone from it, namely each member of the diagram looks at a specific object via a morphism, and all these individual points-of-view must collate nicely in the sense of the commutativity defining the cone. Therefore, a diagram sees an object, typically, in a much more intricate way than any one of its members does (since they somehow need to agree on what they see). It's a bit like collectively a group of people who believe in $B$, see the world differently than the way they do individually. The collective (encoded perhaps the morphisms in the diagram) dictates the behaviour. Back to maths, it may be that the way the diagram perceives the world is identical to the way some object somewhere perceives the world. That object represents the diagram then. It is it's colimit. Limits are similar, but it's more the way the rest of the words sees the diagram (arrows are reversed).
