In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.
I know the definition (several of them), I've read the nlab page (and any good answers will be added there), I've worked with them, I've found examples of functors with and without adjoints, but I couldn't explain what an adjunction is to a five-year-old, the man on the Clapham omnibus, or even an advanced undergraduate.
So how should I intuitively think of adjunctions?
For more background: I'm a topologist by trade who's been learning category theory recently (and, for the most part, enjoying it) but haven't truly internalised it yet. I'm fully convinced of the value of adjunctions, but haven't the same intuition into them as I do for, say, the uniqueness of ordinary cohomology.
Best Answer
The example I would give a five-year old is the following.
Take the category $\mathbb R$ whose objects are the real numbers (or perhaps rational numbers for the five-year old) and a single morphism $x \to y$ whenever $x \leq y$. Let $\mathbb Z$ be the full subcategory consisting of the integers. The inclusion $i : \mathbb Z \to \mathbb R$ has a right and a left adjoint: the former is the floor function, the other is the ceiling function.
I think the five-year old will agree that these are approximations so I would then say that left and right adjoints are just jazzed up versions of approximations.